Year 6 Coordinates, Reflection and Translation: worked examples and questions with answers
By the end of this chapter you will be able to
- Write the coordinates of a point in any of the four quadrants, using negative numbers, across first
- Name the quadrant of a point, and explain why a point on an axis is in no quadrant
- Plot points and draw a shape from its corners, and find the length of a side along a grid line
- Find the missing corner of a rectangle from three of its corners
- Translate a point or a shape on the coordinate plane, and describe a translation in words
- Reflect a point or a shape in the x-axis and in the y-axis
- Explain that a translation or a reflection keeps the size and shape of a shape
You should already know
- Negative numbers, counting through zero on a number line (Year 5, Chapter 1; Year 6, Chapter 1)
- Position and direction on a grid, and coordinates in the first quadrant (Year 4, Chapter 10)
- Translations and reflections in a mirror line on a grid (Year 5, Chapter 10)
- Perimeter and area of rectangles (Year 5, Chapter 8)
Where this leads
- Reflecting shapes in lines other than the axes, and coordinates in more complex shapes (KS3)
- Statistics: pie charts, line graphs and the mean (Year 6, Chapter 12)
National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Describe positions on the full coordinate grid (all four quadrants), Draw and translate simple shapes on the coordinate plane, Reflect shapes in the axes.
Exercise P Do you remember?
Non-calculatorMethod. Start at 3 and count back 7 steps.
- Start at 3. The number line starts at 3.3
- Count back 3 steps to reach 0. Counting back from 3 reaches 0 after 3 steps.3 − 3 = 0
- Count back the other 4 steps. We still have 4 steps to go, and they go below 0, so the answer is negative.0 − 4 = −4
- Write the answer. -4
Check. Add 7 back on: −4 + 7 = 3. ✓
Watch out. Going below 0 gives a negative number. Its minus sign goes in front.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Add the lengths of all four sides.
- Write the lengths of the four sides. A rectangle has two long sides and two short sides.7, 3, 7 and 3
- Add the four sides. The perimeter is the distance all the way round.7 + 3 + 7 + 3 = 20
- Write the answer. The perimeter is in squares, one for each unit of length.20
Check. 2 × (7 + 3) = 20. ✓
Watch out. Count all four sides, not just the two you can see first.
Method. A square has four equal sides, so share the perimeter by 4.
- Write the perimeter. The four sides together make the perimeter.36 cm
- Share the perimeter equally between the four sides. All four sides of a square are the same length.36 ÷ 4 = 9
- Write the answer. Each side is 9 cm long.9 cm
Check. 4 × 9 = 36, so the four sides add up to the perimeter. ✓
Watch out. Divide by 4, not by 2. A square has four sides.
Method. Look at each number: a negative number has a minus sign in front of it.
- Look at 0. Zero is neither positive nor negative.not negative
- Look at 6 and 1.5. These have no minus sign, so they are positive.positive
- Look at -6. The minus sign in front means it is less than zero.negative
- Write the answer. -6
Check. -6 is to the left of 0 on a number line. ✓
Watch out. The minus sign tells you the number is below zero.
Method. Look at the digit after the hundreds.
- Find the hundreds. The hundreds digits of 3 762 are 37 hundreds, so we are between 3 700 and 3 800.3 700 and 3 800
- Look at the tens and ones. The last two digits are 62. That is more than 50, so the number rounds up.62 is more than 50
- Round up. Add one hundred to 3 700.3 700 + 100 = 3 800
- Write the answer. 3 800
Check. 3 762 is 38 short of 3 800 and 62 past 3 700, so 3 800 is the nearer hundred. ✓
Watch out. Look at the number after the digit you are rounding to. 5 or more rounds up.
Positions on the full grid
A grid can be drawn with an x-axis that goes across and a y-axis that goes up and down. The two axes cross at the origin, which is the point (0, 0).
Points on the grid have coordinates, written as a pair (x, y). The first number is the across number. It is positive to the right of the origin and negative to the left. The second number is the up or down number. It is positive above the origin and negative below it.
The grid is split into four quadrants. They go round anticlockwise, starting at the top right.
| Quadrant | Across number | Up or down number | Example |
|---|---|---|---|
| first (top right) | positive | positive | (3, 2) |
| second (top left) | negative | positive | (-4, 3) |
| third (bottom left) | negative | negative | (-2, -3) |
| fourth (bottom right) | positive | negative | (4, -1) |
Key ideas
- Always write the across number first: (x, y).
- The origin is (0, 0). Points on the x-axis have y = 0. Points on the y-axis have x = 0.
- The first quadrant is top right. The quadrants go round anticlockwise: second, third, then fourth.
- A point on an axis is not in any quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point B.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −4
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 3
- Write the coordinates. Always write the across number first, then the up or down number.(-4, 3)
Check. Check on the grid: the point is 4 squares left of the y-axis, and it is 3 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −2, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −3, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.third quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 3 squares left changes the across number. Take away 3.x: 0 − 3 = −3
- Move up or down. Moving 4 squares up changes the up or down number. Add 4.y: 0 + 4 = 4
- Write the new point. Write the across number first, then the up or down number.(-3, 4)
Check. Move back 3 squares right and 4 squares down and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Watch out
Common mistake: writing the up or down number first, so (3, 2) becomes (2, 3). Always go across first, then up or down. Another mistake is to forget the minus sign for a point to the left of the origin, or below it.
Exercise 11A Practice: reading and naming positions
Non-calculatorKey words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point A.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 3
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 2
- Write the coordinates. Always write the across number first, then the up or down number.(3, 2)
Check. Check on the grid: the point is 3 squares right of the y-axis, and it is 2 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point D.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 4
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −1
- Write the coordinates. Always write the across number first, then the up or down number.(4, -1)
Check. Check on the grid: the point is 4 squares right of the y-axis, and it is 1 square down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point E.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 0
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 4
- Write the coordinates. Always write the across number first, then the up or down number.(0, 4)
Check. Check on the grid: the point is on the y-axis, and it is 4 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point F.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −5
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −5
- Write the coordinates. Always write the across number first, then the up or down number.(-5, -5)
Check. Check on the grid: the point is 5 squares left of the y-axis, and it is 5 squares down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = 3, so right of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = 2, so above the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.first quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −4, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = 3, so above the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.second quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −6, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = 2, so above the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.second quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 2 squares right changes the across number. Add 2.x: 0 + 2 = 2
- Move up or down. Moving 5 squares down changes the up or down number. Take away 5.y: 0 − 5 = −5
- Write the new point. Write the across number first, then the up or down number.(2, -5)
Check. Move back 2 squares left and 5 squares up and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 4 squares left changes the across number. Take away 4.x: 0 − 4 = −4
- Move up or down. Moving 1 square up changes the up or down number. Add 1.y: 0 + 1 = 1
- Write the new point. Write the across number first, then the up or down number.(-4, 1)
Check. Move back 4 squares right and 1 square down and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 6 squares left changes the across number. Take away 6.x: 0 − 6 = −6
- There is no move up or down. So the up or down number stays the same.y stays 0
- Write the new point. Write the across number first, then the up or down number.(-6, 0)
Check. Move back 6 squares right and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Compare the signs of each point with the quadrant you need.
- Check the signs of each point. Look at the across number and the up or down number of every point in turn.(3, -4): + across, - up or down (-3, -4): - across, - up or down (-3, 4): - across, + up or down
- Find the point with the right signs. The point (3, -4) has the signs that put it in the fourth quadrant.(3, -4) is in the fourth quadrant
- Write the point. Write the across number first, then the up or down number.(3, -4)
Check. (3, -4) has a positive across number and a negative up or down number. ✓
Watch out. Look at the signs first. Do not choose a point just because its numbers look bigger.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = 2, so right of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −5, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.fourth quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Look at the up or down number first, then the across number.
- Look at the up or down number. It is 0, so the point is not above or below the x-axis.y = 0
- Decide where the point is. A point on an axis is on a line, not inside a quadrant.on the x-axis, so no quadrant
- Write the answer. Sam is not right, because the point is not in the third quadrant.No
Check. The quadrants do not include any point on the axes. ✓
Watch out. A point with 0 in it sits on an axis. It is not in a quadrant.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Third quadrant means both numbers are negative. A point 1 square from both axes has 1 as its size in both numbers.
- Use the quadrant. Both numbers are negative in the third quadrant.x < 0 and y < 0
- Find the across number. The point is 1 square from the y-axis, so its across number is -1.x = -1
- Find the up or down number. The point is 1 square from the x-axis, so its up or down number is -1.y = -1
- Write the answer. Write the across number first.(-1, -1)
Check. (-1, -1) is 1 square left of the y-axis and 1 square below the x-axis, in the third quadrant. ✓
Watch out. Both signs are negative in the third quadrant. Do not make either number positive.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the first point to work out the signs of the second.
- Choose a point in the third quadrant. Both numbers are negative. Take P as (-3, -4) to test the rule.P = (-3, -4)
- Keep the across number. Q has the same across number as P, so it is still -3.x = -3
- Change the up or down number. Q has the opposite up or down number, so -4 becomes 4.y = 4
- Find the quadrant of Q. Q is 3 squares left of the y-axis, and 4 squares above the x-axis.second quadrant
Check. Any point in the third quadrant gives the same answer, because the across number stays negative and the up or down number becomes positive. ✓
Watch out. The across number tells you left or right. Changing the sign of the up or down number moves Q above the x-axis.
Watch out
Common mistake: the sign of each number tells you the direction. A point with a negative across number is to the left of the y-axis, even if the number looks big. Check both signs before you name the quadrant.
Exam tip
In a test, write the coordinates with the across number first, and check each sign with a quick picture of the grid.
Plotting points and drawing shapes
To plot a point, start at the origin. Move across along the x-axis first, then move up or down the y-axis. Put a dot where you stop, and label it with its letter.
To draw a shape, plot each corner, then join the corners in order. A side that lies along a grid line has a length. Use the numbers that change. If the side crosses an axis, the two distances from the axis are added.
A rectangle has four right angles, and its opposite sides are equal. If three corners of a rectangle are known, the fourth corner can be found from the two sides you can see.
Key ideas
- Plot a point across first, then up or down.
- A side along a grid line: subtract the numbers that change, or add the distances from the axis if the side crosses it.
- In a rectangle, opposite sides are equal and every corner is a right angle.
- The fourth corner of a rectangle takes the across number of one corner and the up or down number of another.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same up or down number, so the side runs across.−2 and 3
- Find the length. The two points are on opposite sides of the y-axis, so add how far each one is from the y-axis.2 + 3 = 5
- Write the answer. A length is never negative.5
Check. Count the squares from (-2, 3) to (3, 3) one at a time: there are 5. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same up or down number, so the side runs across.1 and 4
- Find the length. The two points are on the same side of the axis, so take the smaller distance away from the bigger one.4 − 1 = 3
- Write the answer. A length is never negative.3
Check. Count the squares from (1, 1) to (4, 1) one at a time: there are 3. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (4, 2) and (-3, 2) have the same up or down number, so they are joined by a side across the grid.up or down number = 2
- Find the vertical side. (4, 2) and (4, -1) have the same across number, so they are joined by a side up or down the grid.across number = 4
- Find the missing corner. It takes the across number of (-3, 2) and the up or down number of (4, -1).across −3, up or down −1
- Write the answer. The fourth corner completes the rectangle.(-3, -1)
Check. (4, 2), (-3, 2), (-3, -1) and (4, -1) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Watch out
Common mistake: taking the length of a side that crosses an axis by subtracting the two numbers. For example, from -2 to 3 is 2 + 3 = 5 squares, not 3 - 2 = 1 square. Also, a corner you were given is never the missing corner.
Exercise 11B Practice: plotting, lengths and missing corners
Non-calculatorKey words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point A.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −4
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −1
- Write the coordinates. Always write the across number first, then the up or down number.(-4, -1)
Check. Check on the grid: the point is 4 squares left of the y-axis, and it is 1 square down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point B.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 1
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −1
- Write the coordinates. Always write the across number first, then the up or down number.(1, -1)
Check. Check on the grid: the point is 1 square right of the y-axis, and it is 1 square down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point C.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = 1
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 1
- Write the coordinates. Always write the across number first, then the up or down number.(1, 1)
Check. Check on the grid: the point is 1 square right of the y-axis, and it is 1 square up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point D.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −4
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 1
- Write the coordinates. Always write the across number first, then the up or down number.(-4, 1)
Check. Check on the grid: the point is 4 squares left of the y-axis, and it is 1 square up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same up or down number, so the side runs across.−4 and 1
- Find the length. The two points are on opposite sides of the y-axis, so add how far each one is from the y-axis.4 + 1 = 5
- Write the answer. A length is never negative.5
Check. Count the squares from (-4, -1) to (1, -1) one at a time: there are 5. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same across number, so the side runs up or down.−1 and 1
- Find the length. The two points are on opposite sides of the x-axis, so add how far each one is from the x-axis.1 + 1 = 2
- Write the answer. A length is never negative.2
Check. Count the squares from (1, -1) to (1, 1) one at a time: there are 2. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Add the lengths of all four sides.
- Write the lengths of the four sides. A rectangle has two long sides and two short sides.5, 2, 5 and 2
- Add the four sides. The perimeter is the distance all the way round.5 + 2 + 5 + 2 = 14
- Write the answer. The perimeter is in squares, one for each unit of length.14
Check. 2 × (5 + 2) = 14. ✓
Watch out. Count all four sides, not just the two you can see first.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (2, 1) and (7, 1) have the same up or down number, so they are joined by a side across the grid.up or down number = 1
- Find the vertical side. (2, 1) and (2, 5) have the same across number, so they are joined by a side up or down the grid.across number = 2
- Find the missing corner. It takes the across number of (7, 1) and the up or down number of (2, 5).across 7, up or down 5
- Write the answer. The fourth corner completes the rectangle.(7, 5)
Check. (2, 1), (7, 1), (7, 5) and (2, 5) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (-1, -2) and (-6, -2) have the same up or down number, so they are joined by a side across the grid.up or down number = −2
- Find the vertical side. (-1, -2) and (-1, 3) have the same across number, so they are joined by a side up or down the grid.across number = −1
- Find the missing corner. It takes the across number of (-6, -2) and the up or down number of (-1, 3).across −6, up or down 3
- Write the answer. The fourth corner completes the rectangle.(-6, 3)
Check. (-1, -2), (-6, -2), (-6, 3) and (-1, 3) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (1, -2) and (-2, -2) have the same up or down number, so they are joined by a side across the grid.up or down number = −2
- Find the vertical side. (1, -2) and (1, 1) have the same across number, so they are joined by a side up or down the grid.across number = 1
- Find the missing corner. It takes the across number of (-2, -2) and the up or down number of (1, 1).across −2, up or down 1
- Write the answer. The fourth corner completes the rectangle.(-2, 1)
Check. (1, -2), (-2, -2), (-2, 1) and (1, 1) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same across number, so the side runs up or down.−1 and 4
- Find the length. The two points are on opposite sides of the x-axis, so add how far each one is from the x-axis.1 + 4 = 5
- Write the answer. A length is never negative.5
Check. Count the squares from (2, -1) to (2, 4) one at a time: there are 5. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Look at each corner, one at a time, and check the sign of each number.
- Check the first corner. The corner (2, 2) has a positive across number and a positive up number.2 > 0 and 2 > 0
- Check the other corners. The other corners are (5, 2), (5, 6) and (2, 6). They are all positive too.all positive
- Decide. All the corners are in the first quadrant, so the whole rectangle is there.Yes
Check. Positive across numbers are right of the y-axis, and positive up numbers are above the x-axis. ✓
Watch out. A shape is in one quadrant only if every corner is in that quadrant.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Find the width and the height of the rectangle, then multiply them.
- Find the width. The points (-5, 0) and (-2, 0) lie on the same line, so the side is 3 squares long.3 squares
- Find the height. The points (-2, 0) and (-2, 3) lie on the same line, so the side is 3 squares high.3 squares
- Multiply the width by the height. Each row has 3 squares, and there are 3 rows.3 × 3 = 9
- Write the answer. 9 squares
Check. Counting the squares row by row: 3 + 3 + 3 = 9. ✓
Watch out. Count the squares inside the shape, not the corners.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same across number, so the side runs up or down.3 and −4
- Find the length. The two points are on opposite sides of the x-axis, so add how far each one is from the x-axis.3 + 4 = 7
- Write the answer. A length is never negative.7
Check. Count the squares from (-2, 3) to (-2, -4) one at a time: there are 7. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (2, 3) and (-4, 3) have the same up or down number, so they are joined by a side across the grid.up or down number = 3
- Find the vertical side. (2, 3) and (2, -1) have the same across number, so they are joined by a side up or down the grid.across number = 2
- Find the missing corner. It takes the across number of (-4, 3) and the up or down number of (2, -1).across −4, up or down −1
- Write the answer. The fourth corner completes the rectangle.(-4, -1)
Check. (2, 3), (-4, 3), (-4, -1) and (2, -1) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Translating on the coordinate plane
A translation slides a shape without turning or flipping it. Every corner moves the same number of squares in the same direction. To find the new coordinates, change the across number by the across move, and change the up or down number by the up or down move.
Moving right or up adds to a number. Moving left or down takes away from a number. For example, the point (-5, 3) moved 5 squares right and 4 squares down becomes (-5 + 5, 3 - 4) = (0, -1).
To describe a translation, find the change in each number. Say the across move first, then the up or down move.
Key ideas
- A translation keeps the size, the shape and the way the shape faces.
- Across change first: right adds, left takes away. Then up or down: up adds, down takes away.
- To describe a translation, say the across move first, then the up or down move.
- To undo a translation, move the image the opposite way.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of corner A.start = (-5, 3)
- Move across first. Moving 5 squares right changes the across number. Add 5.x: −5 + 5 = 0
- Move up or down. Moving 4 squares down changes the up or down number. Take away 4.y: 3 − 4 = −1
- Write the new point. Write the across number first, then the up or down number.(0, -1)
Check. Move back 5 squares left and 4 squares up and you return to (-5, 3). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Compare where the point started with where it ended, one number at a time.
- Compare the across numbers. Work out how far the across number changed.−3 to 2: 5 squares right
- Compare the up or down numbers. Work out how far the up or down number changed.−4 to −1: 3 squares up
- Write the translation. Say the across move first, then the up or down move.5 squares right and 3 squares up
Check. Start at (-3, -4) and move 5 squares right and 3 squares up: you reach (2, -1). ✓
Watch out. Say the across move first. A move with no across part is not a move to the right or left.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Undo the move: move the image the opposite way.
- Write the move. This is the move that took the point to its image.3 squares left and 4 squares down
- Undo the across move. Take the across move away from the image. This is the opposite move.x: −2 + 3 = 1
- Undo the up or down move. Take the up or down move away from the image.y: 1 + 4 = 5
- Write the original point. Write the across number first, then the up or down number.(1, 5)
Check. Move 3 squares left and 4 squares down from (1, 5) and you reach (-2, 1). ✓
Watch out. Undo a move by doing the opposite. Right becomes left, and up becomes down.
Watch out
Common mistake: moving only some of the corners. Every corner must move the same number of squares. Another mistake is to add a move to the wrong number: a move to the right changes the across number, not the up or down number.
Exercise 11C Practice: moving points and describing moves
Non-calculatorKey words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the point.start = (2, 3)
- Move across first. Moving 4 squares right changes the across number. Add 4.x: 2 + 4 = 6
- Move up or down. Moving 1 square down changes the up or down number. Take away 1.y: 3 − 1 = 2
- Write the new point. Write the across number first, then the up or down number.(6, 2)
Check. Move back 4 squares left and 1 square up and you return to (2, 3). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the point.start = (-3, 5)
- Move across first. Moving 6 squares right changes the across number. Add 6.x: −3 + 6 = 3
- Move up or down. Moving 2 squares down changes the up or down number. Take away 2.y: 5 − 2 = 3
- Write the new point. Write the across number first, then the up or down number.(3, 3)
Check. Move back 6 squares left and 2 squares up and you return to (-3, 5). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the point.start = (-1, -4)
- Move across first. Moving 3 squares left changes the across number. Take away 3.x: −1 − 3 = −4
- Move up or down. Moving 5 squares up changes the up or down number. Add 5.y: −4 + 5 = 1
- Write the new point. Write the across number first, then the up or down number.(-4, 1)
Check. Move back 3 squares right and 5 squares down and you return to (-1, -4). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Compare where the point started with where it ended, one number at a time.
- Compare the across numbers. Work out how far the across number changed.−4 to 1: 5 squares right
- Compare the up or down numbers. Work out how far the up or down number changed.2 to −3: 5 squares down
- Write the translation. Say the across move first, then the up or down move.5 squares right and 5 squares down
Check. Start at (-4, 2) and move 5 squares right and 5 squares down: you reach (1, -3). ✓
Watch out. Say the across move first. A move with no across part is not a move to the right or left.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Compare where the point started with where it ended, one number at a time.
- Compare the across numbers. Work out how far the across number changed.3 to −1: 4 squares left
- Compare the up or down numbers. Work out how far the up or down number changed.−2 to 4: 6 squares up
- Write the translation. Say the across move first, then the up or down move.4 squares left and 6 squares up
Check. Start at (3, -2) and move 4 squares left and 6 squares up: you reach (-1, 4). ✓
Watch out. Say the across move first. A move with no across part is not a move to the right or left.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the point.start = (-5, -5)
- Move across first. Moving 10 squares right changes the across number. Add 10.x: −5 + 10 = 5
- There is no move up or down. So the up or down number stays the same.y stays −5
- Write the new point. Write the across number first, then the up or down number.(5, -5)
Check. Move back 10 squares left and you return to (-5, -5). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the origin.start = (0, 0)
- Move across first. Moving 3 squares left changes the across number. Take away 3.x: 0 − 3 = −3
- Move up or down. Moving 3 squares down changes the up or down number. Take away 3.y: 0 − 3 = −3
- Write the new point. Write the across number first, then the up or down number.(-3, -3)
Check. Move back 3 squares right and 3 squares up and you return to (0, 0). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of corner E.start = (4, -4)
- Move across first. Moving 6 squares left changes the across number. Take away 6.x: 4 − 6 = −2
- Move up or down. Moving 3 squares up changes the up or down number. Add 3.y: −4 + 3 = −1
- Write the new point. Write the across number first, then the up or down number.(-2, -1)
Check. Move back 6 squares right and 3 squares down and you return to (4, -4). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of corner W.start = (-5, -4)
- Move across first. Moving 7 squares right changes the across number. Add 7.x: −5 + 7 = 2
- Move up or down. Moving 5 squares up changes the up or down number. Add 5.y: −4 + 5 = 1
- Write the new point. Write the across number first, then the up or down number.(2, 1)
Check. Move back 7 squares left and 5 squares down and you return to (-5, -4). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the point.start = (2, 2)
- Move across first. Moving 3 squares left changes the across number. Take away 3.x: 2 − 3 = −1
- Move up or down. Moving 3 squares down changes the up or down number. Take away 3.y: 2 − 3 = −1
- Write the new point. Write the across number first, then the up or down number.(-1, -1)
Check. Move back 3 squares right and 3 squares up and you return to (2, 2). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(0, 0)
- Move 1: translate 4 squares right and 2 squares down. Every corner moves the same number of squares, so both numbers change by the move.(0, 0) to (4, -2)
x: 0 + 4 = 4; y: 0 − 2 = −2 - Move 2: translate 6 squares left and 1 square up. Every corner moves the same number of squares, so both numbers change by the move.(4, -2) to (-2, -1)
x: 4 − 6 = −2; y: −2 + 1 = −1 - Write the final point. Write across first, then up or down.(-2, -1)
Check. Each move was checked with the number that changed. The final point is (-2, -1). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Compare where the point started with where it ended, one number at a time.
- Compare the across numbers. Work out how far the across number changed.−6 to 0: 6 squares right
- Compare the up or down numbers. The up or down number is the same, so there is no move up or down.4 to 4: no move
- Write the translation. Say the across move first, then the up or down move.6 squares right
Check. Start at (-6, 4) and move 6 squares right: you reach (0, 4). ✓
Watch out. Say the across move first. A move with no across part is not a move to the right or left.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of corner.start = (-2, 1)
- Move across first. Moving 5 squares right changes the across number. Add 5.x: −2 + 5 = 3
- Move up or down. Moving 3 squares down changes the up or down number. Take away 3.y: 1 − 3 = −2
- Write the new point. Write the across number first, then the up or down number.(3, -2)
Check. Move back 5 squares left and 3 squares up and you return to (-2, 1). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of corner.start = (-2, 4)
- Move across first. Moving 5 squares right changes the across number. Add 5.x: −2 + 5 = 3
- Move up or down. Moving 3 squares down changes the up or down number. Take away 3.y: 4 − 3 = 1
- Write the new point. Write the across number first, then the up or down number.(3, 1)
Check. Move back 5 squares left and 3 squares up and you return to (-2, 4). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Reflecting in the x-axis
A reflection flips a shape over a mirror line. The mirror line in this section is the x-axis, which goes across the grid. Each point and its image are the same distance from the x-axis, on opposite sides.
When a point is reflected in the x-axis, its across number stays the same. Its up or down number changes sign. So (3, 4) becomes (3, -4), and (-2, -5) becomes (-2, 5).
A point that is on the x-axis, with an up or down number of 0, does not move.
Key ideas
- Reflecting in the x-axis keeps the across number and changes the sign of the up or down number.
- The point and its image are the same distance from the x-axis, on opposite sides.
- A point on the x-axis does not move when it is reflected in the x-axis.
- The shape keeps its size and shape. Only its position changes.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(3, 4)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 3
- Flip the up or down number. The point is 4 squares above the x-axis. Its image is the same distance below it.y: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(3, -4)
Check. (3, -4) is 4 squares below the x-axis, the same distance as (3, 4) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-2, -5)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −2
- Flip the up or down number. The point is 5 squares below the x-axis. Its image is the same distance above it.y: −5 becomes 5
- Write the image. Write the across number first, then the up or down number.(-2, 5)
Check. (-2, 5) is 5 squares above the x-axis, the same distance as (-2, -5) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Reflect the image back over the mirror line. Reflecting again gives the original point, because it is the same distance from the mirror line.
- Write the point. Write its across number and its up or down number.(5, -2)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 5
- Flip the up or down number. The point is 2 squares below the x-axis. Its image is the same distance above it.y: −2 becomes 2
- Write the image. Write the across number first, then the up or down number.(5, 2)
Check. (5, 2) is 2 squares above the x-axis, the same distance as (5, -2) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Watch out
Common mistake: changing the across number when reflecting in the x-axis. The point (3, 4) becomes (3, -4), not (-3, 4). Only the sign of the up or down number changes.
Exercise 11D Practice: reflecting points in the x-axis
Non-calculatorKey words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(2, 6)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 2
- Flip the up or down number. The point is 6 squares above the x-axis. Its image is the same distance below it.y: 6 becomes −6
- Write the image. Write the across number first, then the up or down number.(2, -6)
Check. (2, -6) is 6 squares below the x-axis, the same distance as (2, 6) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-4, 3)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −4
- Flip the up or down number. The point is 3 squares above the x-axis. Its image is the same distance below it.y: 3 becomes −3
- Write the image. Write the across number first, then the up or down number.(-4, -3)
Check. (-4, -3) is 3 squares below the x-axis, the same distance as (-4, 3) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(5, -1)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 5
- Flip the up or down number. The point is 1 squares below the x-axis. Its image is the same distance above it.y: −1 becomes 1
- Write the image. Write the across number first, then the up or down number.(5, 1)
Check. (5, 1) is 1 squares above the x-axis, the same distance as (5, -1) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-7, -2)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −7
- Flip the up or down number. The point is 2 squares below the x-axis. Its image is the same distance above it.y: −2 becomes 2
- Write the image. Write the across number first, then the up or down number.(-7, 2)
Check. (-7, 2) is 2 squares above the x-axis, the same distance as (-7, -2) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(1, 3)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 1
- Flip the up or down number. The point is 3 squares above the x-axis. Its image is the same distance below it.y: 3 becomes −3
- Write the image. Write the across number first, then the up or down number.(1, -3)
Check. (1, -3) is 3 squares below the x-axis, the same distance as (1, 3) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(4, 1)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 4
- Flip the up or down number. The point is 1 squares above the x-axis. Its image is the same distance below it.y: 1 becomes −1
- Write the image. Write the across number first, then the up or down number.(4, -1)
Check. (4, -1) is 1 squares below the x-axis, the same distance as (4, 1) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(3, -2)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 3
- Flip the up or down number. The point is 2 squares below the x-axis. Its image is the same distance above it.y: −2 becomes 2
- Write the image. Write the across number first, then the up or down number.(3, 2)
Check. (3, 2) is 2 squares above the x-axis, the same distance as (3, -2) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(4, 5)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 4
- Flip the up or down number. The point is 5 squares above the x-axis. Its image is the same distance below it.y: 5 becomes −5
- Write the image. Write the across number first, then the up or down number.(4, -5)
Check. (4, -5) is 5 squares below the x-axis, the same distance as (4, 5) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(3, -5)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 3
- Flip the up or down number. The point is 5 squares below the x-axis. Its image is the same distance above it.y: −5 becomes 5
- Write the image. Write the across number first, then the up or down number.(3, 5)
Check. (3, 5) is 5 squares above the x-axis, the same distance as (3, -5) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same across number, so the side runs up or down.2 and −2
- Find the length. The two points are on opposite sides of the x-axis, so add how far each one is from the x-axis.2 + 2 = 4
- Write the answer. A length is never negative.4
Check. Count the squares from (-4, 2) to (-4, -2) one at a time: there are 4. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(2, 5)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 2
- Flip the up or down number. The point is 5 squares above the x-axis. Its image is the same distance below it.y: 5 becomes −5
- Write the image. Write the across number first, then the up or down number.(2, -5)
Check. (2, -5) is 5 squares below the x-axis, the same distance as (2, 5) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Think about what the x-axis does to a point that is already on it.
- Look at the point. The point (3, 0) has an up or down number of 0. It is on the x-axis.y = 0
- Reflect it. Changing the sign of 0 gives 0 again, so the image is in the same place.-0 = 0
- Decide. Sam is right, because a point on the mirror line does not move.Yes
Check. (3, 0) reflects to (3, 0). ✓
Watch out. A point on the mirror line is its own image.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, 2)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −1
- Flip the up or down number. The point is 2 squares above the x-axis. Its image is the same distance below it.y: 2 becomes −2
- Write the image. Write the across number first, then the up or down number.(-1, -2)
Check. (-1, -2) is 2 squares below the x-axis, the same distance as (-1, 2) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. P is 5 squares left of the y-axis and 3 squares below the x-axis. Reflect it in the x-axis: the across number stays, and the up or down number changes sign.
- Write the point. Write its across number and its up or down number.(-5, -3)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −5
- Flip the up or down number. The point is 3 squares below the x-axis. Its image is the same distance above it.y: −3 becomes 3
- Write the image. Write the across number first, then the up or down number.(-5, 3)
Check. (-5, 3) is 3 squares above the x-axis, the same distance as (-5, -3) is below it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Reflecting in the y-axis
The mirror line in this section is the y-axis, which goes up and down the grid. Each point and its image are the same distance from the y-axis, on opposite sides.
When a point is reflected in the y-axis, its up or down number stays the same. Its across number changes sign. So (4, 2) becomes (-4, 2), and (-5, -3) becomes (5, -3).
A shape that crosses the y-axis is reflected as well. Reflect each corner, then join the new corners in the same order.
Key ideas
- Reflecting in the y-axis keeps the up or down number and changes the sign of the across number.
- The point and its image are the same distance from the y-axis, on opposite sides.
- A point on the y-axis, with an across number of 0, does not move.
- A reflection keeps the size and shape of the shape.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(4, 2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 2
- Flip the across number. The point is 4 squares right of the y-axis. Its image is the same distance left of it.x: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(-4, 2)
Check. (-4, 2) is 4 squares left of the y-axis, the same distance as (4, 2) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-5, -3)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −3
- Flip the across number. The point is 5 squares left of the y-axis. Its image is the same distance right of it.x: −5 becomes 5
- Write the image. Write the across number first, then the up or down number.(5, -3)
Check. (5, -3) is 5 squares right of the y-axis, the same distance as (-5, -3) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(3, 1)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 1
- Flip the across number. The point is 3 squares right of the y-axis. Its image is the same distance left of it.x: 3 becomes −3
- Write the image. Write the across number first, then the up or down number.(-3, 1)
Check. (-3, 1) is 3 squares left of the y-axis, the same distance as (3, 1) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Watch out
Common mistake: changing the up or down number when reflecting in the y-axis. The point (4, 2) becomes (-4, 2): only the across number changes sign. Also, remember that the image is on the other side of the y-axis, the same distance away.
Exercise 11E Practice: reflecting points in the y-axis
Non-calculatorKey words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(3, 5)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 5
- Flip the across number. The point is 3 squares right of the y-axis. Its image is the same distance left of it.x: 3 becomes −3
- Write the image. Write the across number first, then the up or down number.(-3, 5)
Check. (-3, 5) is 3 squares left of the y-axis, the same distance as (3, 5) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-6, 1)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 1
- Flip the across number. The point is 6 squares left of the y-axis. Its image is the same distance right of it.x: −6 becomes 6
- Write the image. Write the across number first, then the up or down number.(6, 1)
Check. (6, 1) is 6 squares right of the y-axis, the same distance as (-6, 1) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(2, -4)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −4
- Flip the across number. The point is 2 squares right of the y-axis. Its image is the same distance left of it.x: 2 becomes −2
- Write the image. Write the across number first, then the up or down number.(-2, -4)
Check. (-2, -4) is 2 squares left of the y-axis, the same distance as (2, -4) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, -7)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −7
- Flip the across number. The point is 1 squares left of the y-axis. Its image is the same distance right of it.x: −1 becomes 1
- Write the image. Write the across number first, then the up or down number.(1, -7)
Check. (1, -7) is 1 squares right of the y-axis, the same distance as (-1, -7) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(1, 2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 2
- Flip the across number. The point is 1 squares right of the y-axis. Its image is the same distance left of it.x: 1 becomes −1
- Write the image. Write the across number first, then the up or down number.(-1, 2)
Check. (-1, 2) is 1 squares left of the y-axis, the same distance as (1, 2) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-3, 4)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 4
- Flip the across number. The point is 3 squares left of the y-axis. Its image is the same distance right of it.x: −3 becomes 3
- Write the image. Write the across number first, then the up or down number.(3, 4)
Check. (3, 4) is 3 squares right of the y-axis, the same distance as (-3, 4) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(5, -1)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −1
- Flip the across number. The point is 5 squares right of the y-axis. Its image is the same distance left of it.x: 5 becomes −5
- Write the image. Write the across number first, then the up or down number.(-5, -1)
Check. (-5, -1) is 5 squares left of the y-axis, the same distance as (5, -1) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-2, 1)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 1
- Flip the across number. The point is 2 squares left of the y-axis. Its image is the same distance right of it.x: −2 becomes 2
- Write the image. Write the across number first, then the up or down number.(2, 1)
Check. (2, 1) is 2 squares right of the y-axis, the same distance as (-2, 1) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-4, -2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −2
- Flip the across number. The point is 4 squares left of the y-axis. Its image is the same distance right of it.x: −4 becomes 4
- Write the image. Write the across number first, then the up or down number.(4, -2)
Check. (4, -2) is 4 squares right of the y-axis, the same distance as (-4, -2) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(3, -5)
- Move 1: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, -5) to (-3, -5)
x: 3 becomes −3 - Move 2: translate 4 squares left. Every corner moves the same number of squares, so both numbers change by the move.(-3, -5) to (-7, -5)
x: −3 − 4 = −7; y: −5 + 0 = −5 - Write the final point. Write across first, then up or down.(-7, -5)
Check. Each move was checked with the number that changed. The final point is (-7, -5). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-2, 6)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 6
- Flip the across number. The point is 2 squares left of the y-axis. Its image is the same distance right of it.x: −2 becomes 2
- Write the image. Write the across number first, then the up or down number.(2, 6)
Check. (2, 6) is 2 squares right of the y-axis, the same distance as (-2, 6) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Think about what the y-axis does to a point that is already on it.
- Look at the point. The point (0, 4) has an across number of 0. It is on the y-axis.x = 0
- Reflect it. Changing the sign of 0 gives 0 again, so the image is in the same place.-0 = 0
- Decide. Ruby is right, because a point on the mirror line does not move.Yes
Check. (0, 4) reflects to (0, 4). ✓
Watch out. A point on the mirror line is its own image.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Compare the across numbers and the up or down numbers of the point and its image.
- Compare the up or down numbers. Both points are 2 squares up, so the up or down numbers are the same.y = 2 for both
- Compare the across numbers. The across numbers are opposite: -5 and 5.-5 and 5 are opposite
- Decide. The mirror line is halfway between the two points, and it is the y-axis.the y-axis
Check. (-5, 2) and (5, 2) are both 5 squares from the y-axis, on opposite sides. ✓
Watch out. Where the across numbers change sign and the up or down number stays the same, the mirror line is the y-axis.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Work through the two reflections for P = (-3, 2), then name the quadrant of the final point.
- Reflect P in the y-axis. The across number changes sign, and the up or down number stays the same.(-3, 2) becomes (3, 2)
- Reflect Q in the x-axis. The across number stays the same, and the up or down number changes sign.(3, 2) becomes (3, -2)
- Name the quadrant of R. Positive across and negative up or down: right and below the origin.fourth quadrant
Check. Any point in the second quadrant gives the same answer. The final point, (3, -2), is in the fourth quadrant. ✓
Watch out. Two reflections change both signs. Check the sign of each number after every step.
Keeping the shape, and two moves
A translation or a reflection does not change the size or the shape of a shape. Every side keeps the same length, so the image has the same perimeter and the same area as the original.
Sometimes a shape is moved more than once. Do the moves one at a time, in the order they are given. Check the point after each move before you do the next one.
Key ideas
- A translation or a reflection keeps the lengths of the sides, so the perimeter stays the same.
- Do two moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
- Check each step by looking at the numbers that changed.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same up or down number, so the side runs across.−2 and 1
- Find the length. The two points are on opposite sides of the y-axis, so add how far each one is from the y-axis.2 + 1 = 3
- Write the answer. A length is never negative.3
Check. Count the squares from (-2, 1) to (1, 1) one at a time: there are 3. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. The image has the same side lengths, so add them up.
- Find the image of the rectangle. The rectangle has been reflected in the y-axis, so each corner moves to a new place.(-2, 2), (-6, 2), (-6, 4), (-2, 4)
- Find the sides of the image. A reflection or a translation keeps the length of every side, so the sides are the same as before.4 and 2
- Add the four sides. The perimeter is the distance all the way round.4 + 2 + 4 + 2 = 12
- Write the answer. The perimeter is the same as the perimeter of the original rectangle.12
Check. The original rectangle also has perimeter 12. ✓
Watch out. A reflection or a translation does not change the lengths. Only the position changes.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(3, 2)
- Move 1: reflect in the x-axis. The mirror line is the x-axis, so only the up or down number changes sign.(3, 2) to (3, -2)
y: 2 becomes −2 - Move 2: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, -2) to (-3, -2)
x: 3 becomes −3 - Write the final point. Write across first, then up or down.(-3, -2)
Check. Each move was checked with the number that changed. The final point is (-3, -2). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Watch out
Common mistake: thinking that a reflected shape is bigger, or that it has a different perimeter. Only its position changes. Another mistake is to do the two moves in the wrong order.
Exercise 11F Practice: keeping the shape, and moves in order
Non-calculatorKey words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. A translation does not change the length of a side. Only its position changes.
- Look at the original side. The side is 5 squares long.5 squares
- Look at the move. Moving the side up 3 squares slides it. It does not stretch or shrink.slide: length unchanged
- Write the answer. The image of the side is still 5 squares long.5
Check. The image is still 5 squares long, so 5 is the answer. ✓
Watch out. A slide keeps every length the same.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Add the lengths of all four sides.
- Write the lengths of the four sides. A rectangle has two long sides and two short sides.4, 2, 4 and 2
- Add the four sides. The perimeter is the distance all the way round.4 + 2 + 4 + 2 = 12
- Write the answer. The perimeter is in squares, one for each unit of length.12
Check. 2 × (4 + 2) = 12. ✓
Watch out. Count all four sides, not just the two you can see first.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. The image has the same side lengths, so add them up.
- Find the image of the rectangle. The rectangle has been reflected in the x-axis, so each corner moves to a new place.(1, -1), (5, -1), (5, -3), (1, -3)
- Find the sides of the image. A reflection or a translation keeps the length of every side, so the sides are the same as before.4 and 2
- Add the four sides. The perimeter is the distance all the way round.4 + 2 + 4 + 2 = 12
- Write the answer. The perimeter is the same as the perimeter of the original rectangle.12
Check. The original rectangle also has perimeter 12. ✓
Watch out. A reflection or a translation does not change the lengths. Only the position changes.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(2, 4)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 2
- Flip the up or down number. The point is 4 squares above the x-axis. Its image is the same distance below it.y: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(2, -4)
Check. (2, -4) is 4 squares below the x-axis, the same distance as (2, 4) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. Think about what a reflection does to the size of a shape.
- Think about the shape. A reflection flips the shape over a mirror line.flip
- Think about the size. The shape is flipped, not stretched, so every side keeps its length.size: unchanged
- Decide. Sam is wrong. The reflected shape is the same size, only in a different place.No
Check. A reflection keeps every length the same. ✓
Watch out. A reflection moves a shape to the other side. It does not make it bigger.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(-4, 5)
- Move 1: reflect in the x-axis. The mirror line is the x-axis, so only the up or down number changes sign.(-4, 5) to (-4, -5)
y: 5 becomes −5 - Move 2: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(-4, -5) to (4, -5)
x: −4 becomes 4 - Write the final point. Write across first, then up or down.(4, -5)
Check. Each move was checked with the number that changed. The final point is (4, -5). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(3, -1)
- Move 1: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, -1) to (-3, -1)
x: 3 becomes −3 - Move 2: translate 2 squares right. Every corner moves the same number of squares, so both numbers change by the move.(-3, -1) to (-1, -1)
x: −3 + 2 = −1; y: −1 + 0 = −1 - Write the final point. Write across first, then up or down.(-1, -1)
Check. Each move was checked with the number that changed. The final point is (-1, -1). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Compare the across numbers and the up or down numbers of the two points.
- Compare the across numbers. The across number is -2 for both points, so it did not change.x = -2 for both
- Compare the up or down numbers. The up or down numbers are 3 and -3, which are opposite.3 and -3 are opposite
- Decide. Only the up or down number changes sign, so the mirror line is the x-axis.the x-axis
Check. (-2, 3) and (-2, -3) are both 2 across from the y-axis, and 3 squares from the x-axis on opposite sides. ✓
Watch out. Look at which number changes sign: the up or down number means the mirror line is the x-axis.
Key words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. Think about what a translation and a reflection each do to the lengths.
- Look at the translation. Sliding the rectangle 4 squares right keeps its side lengths.slide: lengths unchanged
- Look at the reflection. Reflecting in the y-axis flips the rectangle, and its side lengths stay the same.flip: lengths unchanged
- Decide. Both moves keep every side the same length, so the perimeter does not change.No
Check. The same four sides are added, so the perimeter is the same. ✓
Watch out. Both moves change only where the shape is.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(1, 2)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 1
- Flip the up or down number. The point is 2 squares above the x-axis. Its image is the same distance below it.y: 2 becomes −2
- Write the image. Write the across number first, then the up or down number.(1, -2)
Check. (1, -2) is 2 squares below the x-axis, the same distance as (1, 2) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(-2, 3)
- Move 1: translate 5 squares right. Every corner moves the same number of squares, so both numbers change by the move.(-2, 3) to (3, 3)
x: −2 + 5 = 3; y: 3 + 0 = 3 - Move 2: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, 3) to (-3, 3)
x: 3 becomes −3 - Write the final point. Write across first, then up or down.(-3, 3)
Check. Each move was checked with the number that changed. The final point is (-3, 3). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, 4)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays 4
- Flip the across number. The point is 1 squares left of the y-axis. Its image is the same distance right of it.x: −1 becomes 1
- Write the image. Write the across number first, then the up or down number.(1, 4)
Check. (1, 4) is 1 squares right of the y-axis, the same distance as (-1, 4) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, -2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −2
- Flip the across number. The point is 1 squares left of the y-axis. Its image is the same distance right of it.x: −1 becomes 1
- Write the image. Write the across number first, then the up or down number.(1, -2)
Check. (1, -2) is 1 squares right of the y-axis, the same distance as (-1, -2) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-4, -2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −2
- Flip the across number. The point is 4 squares left of the y-axis. Its image is the same distance right of it.x: −4 becomes 4
- Write the image. Write the across number first, then the up or down number.(4, -2)
Check. (4, -2) is 4 squares right of the y-axis, the same distance as (-4, -2) is left of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Chapter summary
| Skill | Method |
|---|---|
| Coordinates | Write (x, y): across first, then up or down. Positive is right or up, and negative is left or down. |
| Quadrants | First top right, second top left, third bottom left, fourth bottom right. Points on an axis are in no quadrant. |
| Plotting a shape | Plot every corner, then join the corners in order. Count the squares along the grid lines. |
| Missing corner | Use the across number and the up or down number of the two sides you can see. |
| Translation | Add the across move to x and the up or down move to y. Right and up add; left and down take away. |
| Reflection in the x-axis | The across number stays. The up or down number changes sign. |
| Reflection in the y-axis | The up or down number stays. The across number changes sign. |
| Lengths | A translation or a reflection keeps every length, so the shape keeps its size and shape. |
Can you…?
- I can write the coordinates of a point in any of the four quadrants, using negative numbers.
- I can say which quadrant a point is in, and when a point is on an axis.
- I can plot points and draw a shape from its corners on the coordinate plane.
- I can find the length of a side that lies along a grid line, including one that crosses an axis.
- I can translate a point or a shape, and describe a translation in words.
- I can reflect a point or a shape in the x-axis, and in the y-axis.
- I can explain that a translation or a reflection keeps the size and shape of a shape.
Key terms
- Coordinates
- A pair of numbers (x, y) that gives the position of a point on a grid.
- Origin
- The point (0, 0), where the x-axis and the y-axis cross.
- Quadrant
- One of the four parts of the grid, numbered anticlockwise from the top right.
- Translation
- A slide of a shape. Every point moves the same distance in the same direction.
- Reflection
- A flip of a shape over a mirror line. Each point is the same distance from the mirror line, on the other side.
- Mirror line
- The line that a shape is reflected in. In this chapter it is an axis.
- Image
- The new shape or point after a translation or a reflection. The image of A is written A'.
- Axis (plural axes)
- One of the two lines of the grid: the x-axis across, and the y-axis up and down.
Exercise 11Z Check-up
Non-calculatorKey words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point P.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −3
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = 2
- Write the coordinates. Always write the across number first, then the up or down number.(-3, 2)
Check. Check on the grid: the point is 3 squares left of the y-axis, and it is 2 squares up from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −7, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −1, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.third quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of the point.start = (4, -2)
- Move across first. Moving 3 squares left changes the across number. Take away 3.x: 4 − 3 = 1
- Move up or down. Moving 5 squares up changes the up or down number. Add 5.y: −2 + 5 = 3
- Write the new point. Write the across number first, then the up or down number.(1, 3)
Check. Move back 3 squares right and 5 squares down and you return to (4, -2). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-5, 3)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −5
- Flip the up or down number. The point is 3 squares above the x-axis. Its image is the same distance below it.y: 3 becomes −3
- Write the image. Write the across number first, then the up or down number.(-5, -3)
Check. (-5, -3) is 3 squares below the x-axis, the same distance as (-5, 3) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(6, -2)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −2
- Flip the across number. The point is 6 squares right of the y-axis. Its image is the same distance left of it.x: 6 becomes −6
- Write the image. Write the across number first, then the up or down number.(-6, -2)
Check. (-6, -2) is 6 squares left of the y-axis, the same distance as (6, -2) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (2, 1) and (-4, 1) have the same up or down number, so they are joined by a side across the grid.up or down number = 1
- Find the vertical side. (2, 1) and (2, 5) have the same across number, so they are joined by a side up or down the grid.across number = 2
- Find the missing corner. It takes the across number of (-4, 1) and the up or down number of (2, 5).across −4, up or down 5
- Write the answer. The fourth corner completes the rectangle.(-4, 5)
Check. (2, 1), (-4, 1), (-4, 5) and (2, 5) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Compare where the point started with where it ended, one number at a time.
- Compare the across numbers. Work out how far the across number changed.−3 to 2: 5 squares right
- Compare the up or down numbers. Work out how far the up or down number changed.−2 to 4: 6 squares up
- Write the translation. Say the across move first, then the up or down move.5 squares right and 6 squares up
Check. Start at (-3, -2) and move 5 squares right and 6 squares up: you reach (2, 4). ✓
Watch out. Say the across move first. A move with no across part is not a move to the right or left.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Count the squares along the side, using only the numbers that change.
- Find the numbers that change. The two points have the same up or down number, so the side runs across.−6 and −1
- Find the length. The two points are on the same side of the axis, so take the smaller distance away from the bigger one.6 − 1 = 5
- Write the answer. A length is never negative.5
Check. Count the squares from (-6, 3) to (-1, 3) one at a time: there are 5. ✓
Watch out. When the side crosses an axis, add the two distances from the axis. Do not subtract them.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Reflect the image back over the mirror line. Reflecting again gives the original point, because it is the same distance from the mirror line.
- Write the point. Write its across number and its up or down number.(4, -3)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −3
- Flip the across number. The point is 4 squares right of the y-axis. Its image is the same distance left of it.x: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(-4, -3)
Check. (-4, -3) is 4 squares left of the y-axis, the same distance as (4, -3) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(-2, 1)
- Move 1: translate 3 squares right and 2 squares down. Every corner moves the same number of squares, so both numbers change by the move.(-2, 1) to (1, -1)
x: −2 + 3 = 1; y: 1 − 2 = −1 - Move 2: reflect in the x-axis. The mirror line is the x-axis, so only the up or down number changes sign.(1, -1) to (1, 1)
y: −1 becomes 1 - Write the final point. Write across first, then up or down.(1, 1)
Check. Each move was checked with the number that changed. The final point is (1, 1). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Total 22 marks. Allow about 22 minutes.
Exercise 11X SATs-style questions
Non-calculatorKey words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Read the across number first, then the up or down number, from the origin for point T.
- Start at the origin. The origin is (0, 0), where the two axes cross.(0, 0)
- Read the across number first. Move across along the x-axis. Move left if the number is negative.x = −3
- Read the up or down number. Move up along the y-axis. Move down if the number is negative.y = −4
- Write the coordinates. Always write the across number first, then the up or down number.(-3, -4)
Check. Check on the grid: the point is 3 squares left of the y-axis, and it is 4 squares down from the x-axis. ✓
Watch out. Across is always the first number. Do not swap the two numbers.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Compare the signs of each point with the quadrant you need.
- Check the signs of each point. Look at the across number and the up or down number of every point in turn.(3, 4): + across, + up or down (-2, 5): - across, + up or down (-4, -1): - across, - up or down
- Find the point with the right signs. The point (-2, 5) has the signs that put it in the second quadrant.(-2, 5) is in the second quadrant
- Write the point. Write the across number first, then the up or down number.(-2, 5)
Check. (-2, 5) has a negative across number and a positive up or down number. ✓
Watch out. Look at the signs first. Do not choose a point just because its numbers look bigger.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-1, 6)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −1
- Flip the up or down number. The point is 6 squares above the x-axis. Its image is the same distance below it.y: 6 becomes −6
- Write the image. Write the across number first, then the up or down number.(-1, -6)
Check. (-1, -6) is 6 squares below the x-axis, the same distance as (-1, 6) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of corner M.start = (-2, 4)
- Move across first. Moving 4 squares right changes the across number. Add 4.x: −2 + 4 = 2
- Move up or down. Moving 3 squares down changes the up or down number. Take away 3.y: 4 − 3 = 1
- Write the new point. Write the across number first, then the up or down number.(2, 1)
Check. Move back 4 squares left and 3 squares up and you return to (-2, 4). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. The perimeter is the distance all the way round the outside of a shape. A rectangle has four right angles, and its opposite sides are equal.
Method. Use the two sides you can see to find the corner that completes the rectangle.
- Find the horizontal side. (-1, -3) and (-6, -3) have the same up or down number, so they are joined by a side across the grid.up or down number = −3
- Find the vertical side. (-1, -3) and (-1, 2) have the same across number, so they are joined by a side up or down the grid.across number = −1
- Find the missing corner. It takes the across number of (-6, -3) and the up or down number of (-1, 2).across −6, up or down 2
- Write the answer. The fourth corner completes the rectangle.(-6, 2)
Check. (-1, -3), (-6, -3), (-6, 2) and (-1, 2) make a rectangle, with opposite sides the same length. ✓
Watch out. Do not copy one of the corners you were given. The missing corner is a new point.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Do the moves one at a time, in the order they are given.
- Start with the point. Write its coordinates before any move.(3, -2)
- Move 1: reflect in the y-axis. The mirror line is the y-axis, so only the across number changes sign.(3, -2) to (-3, -2)
x: 3 becomes −3 - Move 2: translate 5 squares down. Every corner moves the same number of squares, so both numbers change by the move.(-3, -2) to (-3, -7)
x: −3 + 0 = −3; y: −2 − 5 = −7 - Write the final point. Write across first, then up or down.(-3, -7)
Check. Each move was checked with the number that changed. The final point is (-3, -7). ✓
Watch out. Do the moves in the order they are given. Reflecting and then translating can give a different answer from translating and then reflecting.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Compare where the point started with where it ended, one number at a time.
- Compare the across numbers. Work out how far the across number changed.−4 to 1: 5 squares right
- Compare the up or down numbers. Work out how far the up or down number changed.3 to −2: 5 squares down
- Write the translation. Say the across move first, then the up or down move.5 squares right and 5 squares down
Check. Start at (-4, 3) and move 5 squares right and 5 squares down: you reach (1, -2). ✓
Watch out. Say the across move first. A move with no across part is not a move to the right or left.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of point B.start = (-1, 1)
- Move across first. Moving 5 squares right changes the across number. Add 5.x: −1 + 5 = 4
- Move up or down. Moving 5 squares down changes the up or down number. Take away 5.y: 1 − 5 = −4
- Write the new point. Write the across number first, then the up or down number.(4, -4)
Check. Move back 5 squares left and 5 squares up and you return to (-1, 1). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(4, 3)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays 4
- Flip the up or down number. The point is 3 squares above the x-axis. Its image is the same distance below it.y: 3 becomes −3
- Write the image. Write the across number first, then the up or down number.(4, -3)
Check. (4, -3) is 3 squares below the x-axis, the same distance as (4, 3) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(4, -3)
- Look at the mirror line. The mirror line is the y-axis. The point flips over it, so the up or down number does not change.y stays −3
- Flip the across number. The point is 4 squares right of the y-axis. Its image is the same distance left of it.x: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(-4, -3)
Check. (-4, -3) is 4 squares left of the y-axis, the same distance as (4, -3) is right of it. ✓
Watch out. Reflecting in the y-axis changes the across number, not the up or down number.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = −4, so left of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −3, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.third quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Undo the move: move the image the opposite way.
- Write the move. This is the move that took the point to its image.3 squares left and 4 squares up
- Undo the across move. Take the across move away from the image. This is the opposite move.x: 1 + 3 = 4
- Undo the up or down move. Take the up or down move away from the image.y: −2 − 4 = −6
- Write the original point. Write the across number first, then the up or down number.(4, -6)
Check. Move 3 squares left and 4 squares up from (4, -6) and you reach (1, -2). ✓
Watch out. Undo a move by doing the opposite. Right becomes left, and up becomes down.
Key words. The origin is the point (0, 0), where the x-axis and the y-axis cross. A coordinate is a pair of numbers (x, y). The x-value is the across number, and it is always written first.
Method. Use the signs of the two numbers to find the quadrant.
- Look at the across number. Positive means right of the y-axis. Negative means left of it.x = 4, so right of the y-axis
- Look at the up or down number. Positive means above the x-axis. Negative means below it.y = −6, so below the x-axis
- Name the quadrant. The quadrants go round anticlockwise, starting at the top right.fourth quadrant
Check. The first quadrant is top right, the second is top left, the third is bottom left and the fourth is bottom right. ✓
Watch out. Look at the signs, not the size of the numbers. Two negative signs make the third quadrant.
Key words. A reflection flips a shape over a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides. In this chapter the mirror lines are the axes.
Method. Flip the point over the mirror line. It stays the same distance from the mirror line, on the other side.
- Write the point. Write its across number and its up or down number.(-2, 4)
- Look at the mirror line. The mirror line is the x-axis. The point flips over it, so the across number does not change.x stays −2
- Flip the up or down number. The point is 4 squares above the x-axis. Its image is the same distance below it.y: 4 becomes −4
- Write the image. Write the across number first, then the up or down number.(-2, -4)
Check. (-2, -4) is 4 squares below the x-axis, the same distance as (-2, 4) is above it. ✓
Watch out. Reflecting in the x-axis changes the up or down number, not the across number.
Key words. A translation slides a shape. Every point moves the same distance in the same direction. The new shape is the image. The image of corner A is written A'.
Method. Move the across number first, then the up or down number, one move at a time.
- Start at the point. Write the across number and the up or down number of image of C.start = (-2, -4)
- Move across first. Moving 4 squares right changes the across number. Add 4.x: −2 + 4 = 2
- Move up or down. Moving 1 square up changes the up or down number. Add 1.y: −4 + 1 = −3
- Write the new point. Write the across number first, then the up or down number.(2, -3)
Check. Move back 4 squares left and 1 square down and you return to (-2, -4). ✓
Watch out. Moving left or down takes away from the number. Moving right or up adds to it.
Key words. A translation or a reflection keeps the size and the shape of a shape. Only its position changes.
Method. Look at the sides of the triangle before and after the moves.
- Find the length of AB before the moves. A and B are on the same line, and the distance between them is 2 squares.AB = 2
- Find the length of A'B' after the moves. Reflection and translation keep the lengths, so A'B' is also 2 squares.A'B' = 2
- Explain the shape. Every side keeps its length, and the angles keep their size, so the shape is the same.The triangle has not changed shape or size.
Check. Only the position changes, so the triangle is congruent to the original. ✓
Watch out. A reflection or a translation keeps the size and shape. It only moves the shape.