GCSE Maths: Horizontal and vertical lines, and drawing graphs from tables (Foundation)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Horizontal and vertical lines, and drawing graphs from tables
An equation of a line is a rule that every point on the line obeys. Plot the points (3, −2), (3, 0) and (3, 4). They all have x = 3, and they line up straight up and down. The rule “x is always 3” is written x = 3, and it is the equation of a vertical line that passes through 3 on the x-axis. In the same way y = −2 is a horizontal line passing through −2 on the y-axis, because every point on it has y = −2.
Two special lines are the axes themselves. The y-axis is the line x = 0 and the x-axis is the line y = 0.
Drawing a line from its equation
Most lines are neither flat nor upright. A line such as y = 2x + 1 is a rule that tells you what y is for any x: double the x, then add 1. To draw it:
- Make a table of values. Choose some x values (three or four, including negatives and 0) and work out y for each.
- Plot the points (x, y) on a grid.
- Join them with a ruler to make a straight line, and extend it across the grid. Add the equation as a label.
Why does this give a straight line? Because x appears only on its own (not as x² or 1/x), so each time x goes up by 1, y always goes up by the same amount. Equations like this are called linear. You only need two points to draw a straight line, but plot at least three: if they do not line up, you have made a mistake in your working.
When you substitute a negative number, put it in brackets: if x = −2 in y = 2x + 1 write 2 × (−2) + 1 = −4 + 1 = −3. A point is on the line if its coordinates make the equation true. For example (3, 7) is on y = 2x + 1 because 2 × 3 + 1 = 7, but (3, 8) is not.
Tip: choose x values that give whole-number y values. If the equation has a fraction such as y = ½x + 1, choose even values of x (such as −2, 0, 2, 4) so that the halves disappear.
Key ideas
- x = a is a vertical line; y = b is a horizontal line. The axes are y = 0 (the x-axis) and x = 0 (the y-axis).
- To draw a line: table of values → plot the points → ruler → label.
- Substitute negatives with brackets: 2 × (−3) = −6.
- A point is on a line only if its coordinates make the equation true.
- Check that all your points line up: if one does not, recheck its y value.
Key words. A line is horizontal if it is flat, like the horizon, and vertical if it goes straight up and down. The equation of a line is a rule that every point on the line obeys.
Method. A vertical (up and down) line has one coordinate that never changes. The equation says what that coordinate is.
- Pick three points on the line. Choose points where the line crosses the grid lines so they are easy to read.(2, −2), (2, 1), (2, 3)
- Look for the coordinate that does not change. Every point on a vertical line is directly above or below the others, so they share the same x-coordinate.The x-coordinate is 2 every time. The y-coordinate changes.
- Write the equation. The equation says what is true for every point on the line: x is always 2.x = 2
Check. It crosses the x-axis at (2, 0). ✓
Watch out. Vertical lines have equations x = a number; horizontal lines have equations y = a number. Test one point: (3, 0) is on x = 3 but not on y = 3.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −2: y = 2 × (−2) − 1 = −4 − 1 = −5
x = −1: y = 2 × (−1) − 1 = −2 − 1 = −3
x = 0: y = 2 × 0 − 1 = 0 − 1 = −1
x = 1: y = 2 × 1 − 1 = 2 − 1 = 1
x = 2: y = 2 × 2 − 1 = 4 − 1 = 3
x = 3: y = 2 × 3 − 1 = 6 − 1 = 5x −2 −1 0 1 2 3 y −5 −3 −1 1 3 5 - Plot the points. Each pair is a coordinate: across first, then up or down.(−2, −5) (−1, −3) (0, −1) (1, 1) (2, 3) (3, 5)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −2: y = −(−2) + 3 = 2 + 3 = 5
x = 0: y = −0 + 3 = 0 + 3 = 3
x = 2: y = −2 + 3 = −2 + 3 = 1
x = 4: y = −4 + 3 = −4 + 3 = −1x −2 0 2 4 y 5 3 1 −1 - Plot the points. Each pair is a coordinate: across first, then up or down.(−2, 5) (0, 3) (2, 1) (4, −1)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 3 × 4 − 2 = 10
- Compare with the y-coordinate of the point. The point is (4, 10), so its y-coordinate is 10.10 = 10
- Write the conclusion. The point is on the line only if the two values are equal.(4, 10) is on the line y = 3x − 2.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
Watch out
Vertical or horizontal? x = 3 is vertical (it goes through 3 on the x-axis) and y = 3 is horizontal. If you are unsure, plot (3, 0) and (0, 3) and see which one is on the line. Also: when x is negative, put it in brackets before you multiply, or you may lose the minus sign.Exercise 12D Horizontal and vertical lines, tables of values and drawing lines
Non-calculatorKey words. A line is horizontal if it is flat, like the horizon, and vertical if it goes straight up and down. The equation of a line is a rule that every point on the line obeys.
Method. A vertical (up and down) line has one coordinate that never changes. The equation says what that coordinate is.
- Pick three points on the line. Choose points where the line crosses the grid lines so they are easy to read.(2, −2), (2, 1), (2, 3)
- Look for the coordinate that does not change. Every point on a vertical line is directly above or below the others, so they share the same x-coordinate.The x-coordinate is 2 every time. The y-coordinate changes.
- Write the equation. The equation says what is true for every point on the line: x is always 2.x = 2
Check. It crosses the x-axis at (2, 0). ✓
Watch out. Vertical lines have equations x = a number; horizontal lines have equations y = a number. Test one point: (3, 0) is on x = 3 but not on y = 3.
Key words. A line is horizontal if it is flat, like the horizon, and vertical if it goes straight up and down. The equation of a line is a rule that every point on the line obeys.
Method. A horizontal (flat) line has one coordinate that never changes. The equation says what that coordinate is.
- Pick three points on the line. Choose points where the line crosses the grid lines so they are easy to read.(−3, −3), (0, −3), (4, −3)
- Look for the coordinate that does not change. Every point on a horizontal line is level with the others, so they share the same y-coordinate.The y-coordinate is −3 every time. The x-coordinate changes.
- Write the equation. The equation says what is true for every point on the line: y is always −3.y = −3
Check. It crosses the y-axis at (0, −3). ✓
Watch out. Vertical lines have equations x = a number; horizontal lines have equations y = a number. Test one point: (3, 0) is on x = 3 but not on y = 3.
Key words. A line is horizontal if it is flat, like the horizon, and vertical if it goes straight up and down. The equation of a line is a rule that every point on the line obeys.
Method. A vertical (up and down) line has one coordinate that never changes. The equation says what that coordinate is.
- Pick three points on the line. Choose points where the line crosses the grid lines so they are easy to read.(−4, −2), (−4, 1), (−4, 3)
- Look for the coordinate that does not change. Every point on a vertical line is directly above or below the others, so they share the same x-coordinate.The x-coordinate is −4 every time. The y-coordinate changes.
- Write the equation. The equation says what is true for every point on the line: x is always −4.x = −4
Check. It crosses the x-axis at (−4, 0). ✓
Watch out. Vertical lines have equations x = a number; horizontal lines have equations y = a number. Test one point: (3, 0) is on x = 3 but not on y = 3.
Key words. A line is horizontal if it is flat, like the horizon, and vertical if it goes straight up and down. The equation of a line is a rule that every point on the line obeys.
Method. A horizontal (flat) line has one coordinate that never changes. The equation says what that coordinate is.
- Pick three points on the line. Choose points where the line crosses the grid lines so they are easy to read.(−3, 1), (0, 1), (4, 1)
- Look for the coordinate that does not change. Every point on a horizontal line is level with the others, so they share the same y-coordinate.The y-coordinate is 1 every time. The x-coordinate changes.
- Write the equation. The equation says what is true for every point on the line: y is always 1.y = 1
Check. It crosses the y-axis at (0, 1). ✓
Watch out. Vertical lines have equations x = a number; horizontal lines have equations y = a number. Test one point: (3, 0) is on x = 3 but not on y = 3.
Method. Use the coordinate that stays the same along the line.
- Decide which coordinate stays the same. A vertical line has the same x everywhere; a horizontal line has the same y everywhere.Vertical: x never changes
- Take the x-coordinate of the point. The point is (−2, 4).x = −2
- Write the equation. x = −2
Check. The point (−2, 4) has x = −2, so it is on this line. ✓
Watch out. Do not write the equation with the wrong letter: vertical lines use x.
Method. Use the coordinate that stays the same along the line.
- Decide which coordinate stays the same. A vertical line has the same x everywhere; a horizontal line has the same y everywhere.Horizontal: y never changes
- Take the y-coordinate of the point. The point is (3, 5).y = 5
- Write the equation. y = 5
Check. The point (3, 5) has y = 5, so it is on this line. ✓
Watch out. Do not write the equation with the wrong letter: vertical lines use x.
Method. Use the coordinate that stays the same along the line.
- Decide which coordinate stays the same. A vertical line has the same x everywhere; a horizontal line has the same y everywhere.Horizontal: y never changes
- Take the y-coordinate of the point. The point is (0, 0).y = 0
- Write the equation. y = 0
Check. The point (0, 0) has y = 0, so it is on this line. ✓
Watch out. Do not write the equation with the wrong letter: vertical lines use x.
Method. Use the coordinate that stays the same along the line.
- Decide which coordinate stays the same. A vertical line has the same x everywhere; a horizontal line has the same y everywhere.Vertical: x never changes
- Take the x-coordinate of the point. The point is (0, 0).x = 0
- Write the equation. x = 0
Check. The point (0, 0) has x = 0, so it is on this line. ✓
Watch out. Do not write the equation with the wrong letter: vertical lines use x.
Key words. A table of values lists some x values and the matching y values. To substitute means to replace a letter with a number.
Method. Put each value of x into the equation and work out y. Do the multiplication before the addition.
- Write the rule. The equation tells us how to get y from x: multiply by the gradient, then add the constant.y = x + 1
- Substitute each x value in turn. Replace the letter x with the number. Put negative numbers in brackets so you do not lose the sign.x = 0: y = 0 + 1 = 1
x = 1: y = 1 + 1 = 2
x = 2: y = 2 + 1 = 3
x = 3: y = 3 + 1 = 4
x = 4: y = 4 + 1 = 5 - Write the answers in the table. Each pair (x, y) is a point on the line.
x 0 1 2 3 4 y 1 2 3 4 5
Check. The y-values change by the same amount each time (the gradient 1 times the step in x): 1, 1, 1, 1. ✓
Watch out. Careful with negatives: 2 × (−3) = −6, and −6 + 1 = −5. A negative times a positive is negative.
Key words. A table of values lists some x values and the matching y values. To substitute means to replace a letter with a number.
Method. Put each value of x into the equation and work out y. Do the multiplication before the addition.
- Write the rule. The equation tells us how to get y from x: multiply by the gradient, then add the constant.y = 2x
- Substitute each x value in turn. Replace the letter x with the number. Put negative numbers in brackets so you do not lose the sign.x = −2: y = 2 × (−2) = −4
x = −1: y = 2 × (−1) = −2
x = 0: y = 2 × 0 = 0
x = 1: y = 2 × 1 = 2
x = 2: y = 2 × 2 = 4 - Write the answers in the table. Each pair (x, y) is a point on the line.
x −2 −1 0 1 2 y −4 −2 0 2 4
Check. The y-values change by the same amount each time (the gradient 2 times the step in x): 2, 2, 2, 2. ✓
Watch out. Careful with negatives: 2 × (−3) = −6, and −6 + 1 = −5. A negative times a positive is negative.
Key words. A table of values lists some x values and the matching y values. To substitute means to replace a letter with a number.
Method. Put each value of x into the equation and work out y. Do the multiplication before the addition.
- Write the rule. The equation tells us how to get y from x: multiply by the gradient, then add the constant.y = 3x − 1
- Substitute each x value in turn. Replace the letter x with the number. Put negative numbers in brackets so you do not lose the sign.x = −1: y = 3 × (−1) − 1 = −3 − 1 = −4
x = 0: y = 3 × 0 − 1 = 0 − 1 = −1
x = 1: y = 3 × 1 − 1 = 3 − 1 = 2
x = 2: y = 3 × 2 − 1 = 6 − 1 = 5
x = 3: y = 3 × 3 − 1 = 9 − 1 = 8 - Write the answers in the table. Each pair (x, y) is a point on the line.
x −1 0 1 2 3 y −4 −1 2 5 8
Check. The y-values change by the same amount each time (the gradient 3 times the step in x): 3, 3, 3, 3. ✓
Watch out. Careful with negatives: 2 × (−3) = −6, and −6 + 1 = −5. A negative times a positive is negative.
Key words. A table of values lists some x values and the matching y values. To substitute means to replace a letter with a number.
Method. Put each value of x into the equation and work out y. Do the multiplication before the addition.
- Write the rule. The equation tells us how to get y from x: multiply by the gradient, then add the constant.y = −2x + 10
- Substitute each x value in turn. Replace the letter x with the number. Put negative numbers in brackets so you do not lose the sign.x = 0: y = (−2) × 0 + 10 = 0 + 10 = 10
x = 1: y = (−2) × 1 + 10 = −2 + 10 = 8
x = 2: y = (−2) × 2 + 10 = −4 + 10 = 6
x = 3: y = (−2) × 3 + 10 = −6 + 10 = 4
x = 4: y = (−2) × 4 + 10 = −8 + 10 = 2 - Write the answers in the table. Each pair (x, y) is a point on the line.
x 0 1 2 3 4 y 10 8 6 4 2
Check. The y-values change by the same amount each time (the gradient −2 times the step in x): −2, −2, −2, −2. ✓
Watch out. Careful with negatives: 2 × (−3) = −6, and −6 + 1 = −5. A negative times a positive is negative.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −2: y = 2 × (−2) + 1 = −4 + 1 = −3
x = −1: y = 2 × (−1) + 1 = −2 + 1 = −1
x = 0: y = 2 × 0 + 1 = 0 + 1 = 1
x = 1: y = 2 × 1 + 1 = 2 + 1 = 3
x = 2: y = 2 × 2 + 1 = 4 + 1 = 5
x = 3: y = 2 × 3 + 1 = 6 + 1 = 7x −2 −1 0 1 2 3 y −3 −1 1 3 5 7 - Plot the points. Each pair is a coordinate: across first, then up or down.(−2, −3) (−1, −1) (0, 1) (1, 3) (2, 5) (3, 7)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −3: y = (−3) − 2 = −5
x = −2: y = (−2) − 2 = −4
x = −1: y = (−1) − 2 = −3
x = 0: y = 0 − 2 = −2
x = 1: y = 1 − 2 = −1
x = 2: y = 2 − 2 = 0
x = 3: y = 3 − 2 = 1
x = 4: y = 4 − 2 = 2x −3 −2 −1 0 1 2 3 4 y −5 −4 −3 −2 −1 0 1 2 - Plot the points. Each pair is a coordinate: across first, then up or down.(−3, −5) (−2, −4) (−1, −3) (0, −2) (1, −1) (2, 0) (3, 1) (4, 2)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −2: y = −(−2) + 3 = 2 + 3 = 5
x = −1: y = −(−1) + 3 = 1 + 3 = 4
x = 0: y = −0 + 3 = 0 + 3 = 3
x = 1: y = −1 + 3 = −1 + 3 = 2
x = 2: y = −2 + 3 = −2 + 3 = 1
x = 3: y = −3 + 3 = −3 + 3 = 0
x = 4: y = −4 + 3 = −4 + 3 = −1x −2 −1 0 1 2 3 4 y 5 4 3 2 1 0 −1 - Plot the points. Each pair is a coordinate: across first, then up or down.(−2, 5) (−1, 4) (0, 3) (1, 2) (2, 1) (3, 0) (4, −1)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −1: y = 3 × (−1) − 2 = −3 − 2 = −5
x = 0: y = 3 × 0 − 2 = 0 − 2 = −2
x = 1: y = 3 × 1 − 2 = 3 − 2 = 1
x = 2: y = 3 × 2 − 2 = 6 − 2 = 4
x = 3: y = 3 × 3 − 2 = 9 − 2 = 7x −1 0 1 2 3 y −5 −2 1 4 7 - Plot the points. Each pair is a coordinate: across first, then up or down.(−1, −5) (0, −2) (1, 1) (2, 4) (3, 7)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −4: y = 12 × (−4) + 2 = −2 + 2 = 0
x = −2: y = 12 × (−2) + 2 = −1 + 2 = 1
x = 0: y = 12 × 0 + 2 = 0 + 2 = 2
x = 2: y = 12 × 2 + 2 = 1 + 2 = 3
x = 4: y = 12 × 4 + 2 = 2 + 2 = 4x −4 −2 0 2 4 y 0 1 2 3 4 - Plot the points. Each pair is a coordinate: across first, then up or down.(−4, 0) (−2, 1) (0, 2) (2, 3) (4, 4)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Draw a table of values, plot the points, then join them with a straight line.
- Make a table of values. Two points are enough to draw a straight line, but three or more let you spot a mistake: all the points must line up.
- Work out y for each x. Substitute each x into the equation.x = −3: y = (−3) = −3
x = −1: y = (−1) = −1
x = 0: y = 0 = 0
x = 2: y = 2 = 2
x = 3: y = 3 = 3x −3 −1 0 2 3 y −3 −1 0 2 3 - Plot the points. Each pair is a coordinate: across first, then up or down.(−3, −3) (−1, −1) (0, 0) (2, 2) (3, 3)
- Join them with a ruler and extend the line. The graph of a linear equation is a straight line, so the points must line up. If one is out of line, check its working.
Check. All the plotted points lie on one straight line. A point taken from the line, such as one with a different x, also satisfies the equation. ✓
Watch out. Use a ruler and extend the line across the whole grid. If the points do not line up, one y value is wrong.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 2 × 4 − 3 = 5
- Compare with the y-coordinate of the point. The point is (4, 5), so its y-coordinate is 5.5 = 5
- Write the conclusion. The point is on the line only if the two values are equal.(4, 5) is on the line y = 2x − 3.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 2 × (−1) − 3 = −5
- Compare with the y-coordinate of the point. The point is (−1, −5), so its y-coordinate is −5.−5 = −5
- Write the conclusion. The point is on the line only if the two values are equal.(−1, −5) is on the line y = 2x − 3.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 2 × 0 − 3 = −3
- Compare with the y-coordinate of the point. The point is (0, 3), so its y-coordinate is 3.−3 ≠ 3
- Write the conclusion. The point is on the line only if the two values are equal.(0, 3) is not on the line y = 2x − 3.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. A point is on a line if its coordinates make the equation true. Substitute them in.
- Substitute the x-coordinate into the equation. If the point is on the line, the equation gives its y-coordinate.y = 2 × 3 − 3 = 3
- Compare with the y-coordinate of the point. The point is (3, 2), so its y-coordinate is 2.3 ≠ 2
- Write the conclusion. The point is on the line only if the two values are equal.(3, 2) is not on the line y = 2x − 3.
Check. Try the substitution the other way: put the y-coordinate in and see if you get the x-coordinate. ✓
Watch out. Put the x in the x place and the y in the y place.
Method. Substitute into the equation: for part (a) put in x = 4; for part (b) put in y = 11 and solve.
- Part (a): substitute x = 4. Replace x by 4 and work out y.y = 3 × 4 + 2 = 12 + 2 = 14
- Part (b): substitute y = 11. Now y is known, so we have an equation to solve for x.11 = 3x + 2
- Subtract 2 from both sides. This removes the + 2.9 = 3x
- Divide both sides by 3. This leaves x on its own.x = 9 ÷ 3 = 3
Check. Check (b): when x = 3, y = 3 × 3 + 2 = 11. ✓
Watch out. In part (b) you are finding x, so solve an equation. Do not just multiply by 3.
Method. Test each equation with the given points: a correct equation must work for every point.
- Test y = x − 1 with (1, 1). Put x = 1 in. The result must be 1.1 − 1 = 0, but y should be 1. It fails.
- Test y = 2x − 1 with (0, −1), (1, 1), (2, 3). Put each x in and compare with y.2 × 0 − 1 = −1 ✓
2 × 1 − 1 = 1 ✓
2 × 2 − 1 = 3 ✓ - Test y = 2x + 1 with (0, −1). Put x = 0 in.2 × 0 + 1 = 1, but y should be −1. It fails.
- Choose. Only the second equation works for all three points.y = 2x − 1
Check. Also, y goes up by 2 each time x goes up by 1 (−1, 1, 3), which fits 2x, and y = −1 when x = 0 fits − 1. ✓
Watch out. One point is not enough: check at least two points.
Method. Look at the points on the line: for a horizontal line the y-coordinate never changes.
- Write three points with y = 3. Choose any x you like.(−2, 3), (0, 3), (4, 3)
- Look at where they are. They are all at the same height, 3 units above the x-axis, so they lie on a flat line.The line is horizontal.
- State the conclusion. A vertical line would be x = 3.Sam is wrong: y = 3 is a horizontal line, and x = 3 is the vertical one.
Check. Plot (0, 3) and (4, 3): the line joining them is flat. ✓
Watch out. The number is where the line crosses the axis of the same letter: y = 3 crosses the y-axis at 3.
Method. Find the four corners where the lines cross, then use subtraction for the side lengths.
- Find the corners. A vertical line x = a meets a horizontal line y = b at (a, b).(−2, −1), (3, −1), (3, 3), (−2, 3)
- Find the width and the height. Subtract the x-values and the y-values.width = 3 − (−2) = 5, height = 3 − (−1) = 4
- Work out the perimeter and the area. Perimeter = 2 × (5 + 4); area = 5 × 4.perimeter = 18, area = 20
- Draw it. The sketch confirms the rectangle.
Check. 5 × 4 = 20 and 2 × 5 + 2 × 4 = 18. ✓
Watch out. Perimeter and area are different calculations.
| x | 0 | 1 | 2 | 3 |
| y | 1 | 4 | 7 | 11 |
Method. Check whether y increases by the same amount every time x increases by 1. On a straight line it must.
- Write the y values and find the differences. For x going 0, 1, 2, 3 the differences in y are found by subtracting each value from the next.1, 4, 7, 11 → 4 − 1 = 3, 7 − 4 = 3, 11 − 7 = 4
- Spot the odd one. The first two differences are 3, so the pattern is “add 3”. The last difference is 4, so the last value is wrong.The value 11 is wrong.
- Correct it. Add 3 to the previous value.7 + 3 = 10
- Write the rule. It starts at 1 when x = 0 and goes up by 3 each time.y = 3x + 1
Check. Check: 3 × 3 + 1 = 10. ✓
Watch out. On a straight line the y values change by the same amount when x goes up in equal steps.