GCSE Maths: Sketching quadratics and completing the square (Higher)

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² + 4x + 1
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   4 ÷ 2 = 2, so the bracket is (x + 2)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x + 2)² = x² + 4x + 4
4. Take away the extra number and add the original constant. We have an extra 4, so subtract it to keep the value the same. Then put back the constant term.
   x² + 4x + 1 = (x + 2)² − 4 + 1
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x + 2)² − 3

Check. Expand the answer: (x + 2)² − 3 = x² + 4x + 1, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² − 6x + 2
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   −6 ÷ 2 = −3, so the bracket is (x − 3)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x − 3)² = x² − 6x + 9
4. Take away the extra number and add the original constant. We have an extra 9, so subtract it to keep the value the same. Then put back the constant term.
   x² − 6x + 2 = (x − 3)² − 9 + 2
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x − 3)² − 7

Check. Expand the answer: (x − 3)² − 7 = x² − 6x + 2, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² + 10x
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   10 ÷ 2 = 5, so the bracket is (x + 5)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x + 5)² = x² + 10x + 25
4. Take away the extra number and add the original constant. We have an extra 25, so subtract it to keep the value the same. Then put back the constant term.
   x² + 10x = (x + 5)² − 25
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x + 5)² − 25

Check. Expand the answer: (x + 5)² − 25 = x² + 10x, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² − 2x − 5
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   −2 ÷ 2 = −1, so the bracket is (x − 1)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x − 1)² = x² − 2x + 1
4. Take away the extra number and add the original constant. We have an extra 1, so subtract it to keep the value the same. Then put back the constant term.
   x² − 2x − 5 = (x − 1)² − 1 − 5
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x − 1)² − 6

Check. Expand the answer: (x − 1)² − 6 = x² − 2x − 5, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² + 3x
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   3 ÷ 2 = 1.5, so the bracket is (x + 1.5)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x + 1.5)² = x² + 3x + 2.25
4. Take away the extra number and add the original constant. We have an extra 2.25, so subtract it to keep the value the same. Then put back the constant term.
   x² + 3x = (x + 1.5)² − 2.25
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x + 1.5)² − 2.25

Check. Expand the answer: (x + 1.5)² − 2.25 = x² + 3x, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² − 8x + 20
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   −8 ÷ 2 = −4, so the bracket is (x − 4)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x − 4)² = x² − 8x + 16
4. Take away the extra number and add the original constant. We have an extra 16, so subtract it to keep the value the same. Then put back the constant term.
   x² − 8x + 20 = (x − 4)² − 16 + 20
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x − 4)² + 4

Check. Expand the answer: (x − 4)² + 4 = x² − 8x + 20, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² − 4x + 7
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   −4 ÷ 2 = −2, so the bracket is (x − 2)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x − 2)² = x² − 4x + 4
4. Take away the extra number and add the original constant. We have an extra 4, so subtract it to keep the value the same. Then put back the constant term.
   x² − 4x + 7 = (x − 2)² − 4 + 7
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x − 2)² + 3
6. Find the turning point. A squared bracket is never negative, and it equals 0 when the bracket is 0. So y is at its minimum when (x − 2) = 0, that is when x = 2. Then y = 3.
   Turning point = (2, 3)   (minimum value of y is 3)

Check. Expand the answer: (x − 2)² + 3 = x² − 4x + 7, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² + 6x + 5
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   6 ÷ 2 = 3, so the bracket is (x + 3)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x + 3)² = x² + 6x + 9
4. Take away the extra number and add the original constant. We have an extra 9, so subtract it to keep the value the same. Then put back the constant term.
   x² + 6x + 5 = (x + 3)² − 9 + 5
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x + 3)² − 4
6. Find the turning point. A squared bracket is never negative, and it equals 0 when the bracket is 0. So y is at its minimum when (x + 3) = 0, that is when x = −3. Then y = −4.
   Turning point = (−3, −4)   (minimum value of y is −4)

Check. Expand the answer: (x + 3)² − 4 = x² + 6x + 5, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² − 10x + 30
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   −10 ÷ 2 = −5, so the bracket is (x − 5)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x − 5)² = x² − 10x + 25
4. Take away the extra number and add the original constant. We have an extra 25, so subtract it to keep the value the same. Then put back the constant term.
   x² − 10x + 30 = (x − 5)² − 25 + 30
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x − 5)² + 5
6. Find the turning point. A squared bracket is never negative, and it equals 0 when the bracket is 0. So y is at its minimum when (x − 5) = 0, that is when x = 5. Then y = 5.
   Turning point = (5, 5)   (minimum value of y is 5)

Check. Expand the answer: (x − 5)² + 5 = x² − 10x + 30, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. Completing the square means writing x² + bx + c as (x + p)² + q. The turning point is the lowest point (minimum) of a U-shape or the highest point (maximum) of an n-shape.

Method. Complete the square: write the quadratic as a(x + p)² + q. Then the turning point is (−p, q).

1. Write the quadratic and look at the numbers. We want it in the form a(x + p)² + q. That form shows the turning point at a glance.
   y = x² + x + 1
2. Halve the number in front of x. In (x + p)² = x² + 2px + p², the middle term is 2p. So p is half of the number in front of x.
   1 ÷ 2 = 0.5, so the bracket is (x + 0.5)
3. Square the bracket and see what it gives. Expanding the bracket gives the x² and x terms we want, but it also adds an extra number p².
   (x + 0.5)² = x² + 1x + 0.25
4. Take away the extra number and add the original constant. We have an extra 0.25, so subtract it to keep the value the same. Then put back the constant term.
   x² + x + 1 = (x + 0.5)² − 0.25 + 1
5. Tidy up the numbers. Combine the two plain numbers.
   y = (x + 0.5)² + 0.75
6. Find the turning point. A squared bracket is never negative, and it equals 0 when the bracket is 0. So y is at its minimum when (x + 0.5) = 0, that is when x = −0.5. Then y = 0.75.
   Turning point = (−0.5, 0.75)   (minimum value of y is 0.75)

Check. Expand the answer: (x + 0.5)² + 0.75 = x² + x + 1, which is the original. ✓

Watch out. The turning point has the opposite sign to the number in the bracket: (x − 3)² + 1 has its turning point at x = +3.

Key words. To sketch a graph means to draw its shape with the key points labelled, without plotting every point. The intercepts are where the graph crosses the axes.

Method. Find the shape, the y-intercept, the roots and the turning point, then draw a smooth curve through them.

1. Decide the shape. The sign of the x² term tells us: positive gives a U-shape, negative gives an n-shape.
   The number in front of x² is 1, so the curve is U-shaped.
2. Find the y-intercept. The curve crosses the y-axis where x = 0, which is the constant term.
   x = 0:  y = 0² − 4 × 0 + 3 = 0 + 0 + 3 = 3
   The y-intercept is (0, 3).
3. Find the roots. The roots are where the curve meets the x-axis, so put y = 0 and solve. Factorise, then use "if two brackets multiply to 0, one of them is 0".
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = -4 and c = 3.
   We need two numbers that multiply to 3 (the last number) and add to -4 (the middle number). List the pairs that multiply to 3:
   1 × 3 = 3,   add: 1 + 3 = 4 ✗
   -1 × -3 = 3,   add: -1 + -3 = -4 ✓ this pair works
   Put the pair into two brackets: (x − 1)(x − 3).
   Check by expanding: x² − 3x − 1x + 3 = x² − 4x + 3. ✓
   Set (x − 3)(x − 1) = 0, so x = 1 or x = 3
   The curve crosses the x-axis at (1, 0) and (3, 0).
4. Find the turning point. The turning point is on the line of symmetry. Complete the square, or use the midpoint of the roots.
   y = (x − 2)² − 1
   Turning point = (2, −1)
5. Draw the axes and the curve. A sketch does not need an accurate scale. It must show the right shape and label the key points: the y-intercept, any roots and the turning point.
   Label (0, 3), the roots, and the turning point (2, −1).

Check. The curve is symmetrical: the turning point lies halfway between the two roots, and the y-intercept is on the curve (substitute x = 0 to see). ✓

Watch out. A sketch needs labelled coordinates, not a perfect drawing. Make sure the curve is smooth and symmetrical.

Key words. To sketch a graph means to draw its shape with the key points labelled, without plotting every point. The intercepts are where the graph crosses the axes.

Method. Find the shape, the y-intercept, the roots and the turning point, then draw a smooth curve through them.

1. Decide the shape. The sign of the x² term tells us: positive gives a U-shape, negative gives an n-shape.
   The number in front of x² is 1, so the curve is U-shaped.
2. Find the y-intercept. The curve crosses the y-axis where x = 0, which is the constant term.
   x = 0:  y = 0² + 2 × 0 − 8 = 0 + 0 − 8 = −8
   The y-intercept is (0, −8).
3. Find the roots. The roots are where the curve meets the x-axis, so put y = 0 and solve. Factorise, then use "if two brackets multiply to 0, one of them is 0".
   Spot the pattern. This is a quadratic of the form x² + bx + c, with b = 2 and c = -8.
   We need two numbers that multiply to -8 (the last number) and add to 2 (the middle number). List the pairs that multiply to -8:
   1 × -8 = -8,   add: 1 + -8 = -7 ✗
   -1 × 8 = -8,   add: -1 + 8 = 7 ✗
   2 × -4 = -8,   add: 2 + -4 = -2 ✗
   -2 × 4 = -8,   add: -2 + 4 = 2 ✓ this pair works
   Put the pair into two brackets: (x − 2)(x + 4).
   Check by expanding: x² + 4x − 2x − 8 = x² + 2x − 8. ✓
   Set (x − 2)(x + 4) = 0, so x = −4 or x = 2
   The curve crosses the x-axis at (−4, 0) and (2, 0).
4. Find the turning point. The turning point is on the line of symmetry. Complete the square, or use the midpoint of the roots.
   y = (x + 1)² − 9
   Turning point = (−1, −9)
5. Draw the axes and the curve. A sketch does not need an accurate scale. It must show the right shape and label the key points: the y-intercept, any roots and the turning point.
   Label (0, −8), the roots, and the turning point (−1, −9).

Check. The curve is symmetrical: the turning point lies halfway between the two roots, and the y-intercept is on the curve (substitute x = 0 to see). ✓

Watch out. A sketch needs labelled coordinates, not a perfect drawing. Make sure the curve is smooth and symmetrical.

Key words. To sketch a graph means to draw its shape with the key points labelled, without plotting every point. The intercepts are where the graph crosses the axes.

Method. Find the shape, the y-intercept, the roots and the turning point, then draw a smooth curve through them.

1. Decide the shape. The sign of the x² term tells us: positive gives a U-shape, negative gives an n-shape.
   The number in front of x² is −1, so the curve is n-shaped.
2. Find the y-intercept. The curve crosses the y-axis where x = 0, which is the constant term.
   x = 0:  y = −(0)² + 4 = 0 + 4 = 4
   The y-intercept is (0, 4).
3. Find the roots. The roots are where the curve meets the x-axis, so put y = 0 and solve. Factorise, then use "if two brackets multiply to 0, one of them is 0".
   −x² + 4 = −(x − 2)(x + 2)
   Set −(x − 2)(x + 2) = 0, so x = −2 or x = 2
   The curve crosses the x-axis at (−2, 0) and (2, 0).
4. Find the turning point. The turning point is on the line of symmetry. Complete the square, or use the midpoint of the roots.
   y = −(x + 0)² + 4
   Turning point = (0, 4)
5. Draw the axes and the curve. A sketch does not need an accurate scale. It must show the right shape and label the key points: the y-intercept, any roots and the turning point.
   Label (0, 4), the roots, and the turning point (0, 4).

Check. The curve is symmetrical: the turning point lies halfway between the two roots, and the y-intercept is on the curve (substitute x = 0 to see). ✓

Watch out. A sketch needs labelled coordinates, not a perfect drawing. Make sure the curve is smooth and symmetrical.

Key words. To sketch a graph means to draw its shape with the key points labelled, without plotting every point. The intercepts are where the graph crosses the axes.

Method. Find the shape, the y-intercept, the roots and the turning point, then draw a smooth curve through them.

1. Decide the shape. The sign of the x² term tells us: positive gives a U-shape, negative gives an n-shape.
   The number in front of x² is 1, so the curve is U-shaped.
2. Find the y-intercept. The curve crosses the y-axis where x = 0, which is the constant term.
   x = 0:  y = 0² − 6 × 0 + 9 = 0 + 0 + 9 = 9
   The y-intercept is (0, 9).
3. Find the roots. The roots are where y = 0. Here it factorises as a perfect square, so there is one repeated root and the curve just touches the x-axis.
   y = 0: Put the pair into two brackets: (x − 3)(x − 3).
   Check by expanding: x² − 3x − 3x + 9 = x² − 6x + 9. ✓
   x = 3
4. Find the turning point. Complete the square to find it.
   y = (x − 3)²
   Turning point = (3, 0)
5. Draw the axes and the curve. A sketch does not need an accurate scale. It must show the right shape and label the key points: the y-intercept, any roots and the turning point.
   Label (0, 9), the roots, and the turning point (3, 0).

Check. Substitute x = 0 and the x-value of the turning point into the rule to confirm both points lie on the curve. ✓

Watch out. A sketch needs labelled coordinates, not a perfect drawing. Make sure the curve is smooth and symmetrical.

Key words. To sketch a graph means to draw its shape with the key points labelled, without plotting every point. The intercepts are where the graph crosses the axes.

Method. Find the shape, the y-intercept, the roots and the turning point, then draw a smooth curve through them.

1. Decide the shape. The sign of the x² term tells us: positive gives a U-shape, negative gives an n-shape.
   The number in front of x² is 1, so the curve is U-shaped.
2. Find the y-intercept. The curve crosses the y-axis where x = 0, which is the constant term.
   x = 0:  y = 0² + 4 × 0 + 7 = 0 + 0 + 7 = 7
   The y-intercept is (0, 7).
3. Check whether there are roots. The curve has its turning point above the x-axis and opens away from it, so it never meets the x-axis.
   Turning point y-value 3 is positive, so there are no roots.
4. Find the turning point. Complete the square to find it.
   y = (x + 2)² + 3
   Turning point = (−2, 3)
5. Draw the axes and the curve. A sketch does not need an accurate scale. It must show the right shape and label the key points: the y-intercept, any roots and the turning point.
   Label (0, 7), and the turning point (−2, 3).

Check. Substitute x = 0 and the x-value of the turning point into the rule to confirm both points lie on the curve. ✓

Watch out. A sketch needs labelled coordinates, not a perfect drawing. Make sure the curve is smooth and symmetrical.

Key words. A surd is a root that does not simplify to a whole number or fraction, such as √5. Exact means leaving it as √5 instead of a rounded decimal.

Method. Complete the square, then undo it: square root both sides (positive and negative).

1. Put y = 0. Roots are where the curve meets the x-axis.
   0 = x² − 4x + 1
2. Complete the square. Halve the number in front of x (−4 ÷ 2 = −2), square the bracket, and correct the constant.
   0 = (x − 2)² − 3
3. Get the bracket on its own. Move the 3 to the other side (do the inverse operation).
   (x − 2)² = 3
4. Take the square root of both sides. There are two square roots: positive and negative. Leave the answer as a surd (an exact root), simplified if possible.
   x − 2 = ±√3
5. Get x on its own. Do the inverse of subtracting 2.
   x = 2 ± √3

Check. Decimals: 2 + 1.73 = 3.73 and 2 − 1.73 = 0.27. Substituting 3.73 into the rule gives about 0. ✓

Watch out. Do not forget the ±. Leave surds exact unless the question asks for decimals.

Key words. A surd is a root that does not simplify to a whole number or fraction, such as √5. Exact means leaving it as √5 instead of a rounded decimal.

Method. Complete the square, then undo it: square root both sides (positive and negative).

1. Put y = 0. Roots are where the curve meets the x-axis.
   0 = x² + 6x + 2
2. Complete the square. Halve the number in front of x (6 ÷ 2 = 3), square the bracket, and correct the constant.
   0 = (x + 3)² − 7
3. Get the bracket on its own. Move the 7 to the other side (do the inverse operation).
   (x + 3)² = 7
4. Take the square root of both sides. There are two square roots: positive and negative. Leave the answer as a surd (an exact root), simplified if possible.
   x + 3 = ±√7
5. Get x on its own. Do the inverse of adding 3.
   x = −3 ± √7

Check. Decimals: −3 + 2.65 = −0.35 and −3 − 2.65 = −5.65. Substituting −0.35 into the rule gives about 0. ✓

Watch out. Do not forget the ±. Leave surds exact unless the question asks for decimals.

Key words. A surd is a root that does not simplify to a whole number or fraction, such as √5. Exact means leaving it as √5 instead of a rounded decimal.

Method. Complete the square, then undo it: square root both sides (positive and negative).

1. Put y = 0. Roots are where the curve meets the x-axis.
   0 = x² − 2x − 5
2. Complete the square. Halve the number in front of x (−2 ÷ 2 = −1), square the bracket, and correct the constant.
   0 = (x − 1)² − 6
3. Get the bracket on its own. Move the 6 to the other side (do the inverse operation).
   (x − 1)² = 6
4. Take the square root of both sides. There are two square roots: positive and negative. Leave the answer as a surd (an exact root), simplified if possible.
   x − 1 = ±√6
5. Get x on its own. Do the inverse of subtracting 1.
   x = 1 ± √6

Check. Decimals: 1 + 2.45 = 3.45 and 1 − 2.45 = −1.45. Substituting 3.45 into the rule gives about 0. ✓

Watch out. Do not forget the ±. Leave surds exact unless the question asks for decimals.

Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).

Method. The turning point of y = a(x + p)² + q is (−p, q). The bracket gives the x-value (opposite sign) and the number on the end gives the y-value.

1. Find the x-value. It makes the bracket equal to 0, so it has the opposite sign to the number inside the bracket.
   x = 5
2. Find the y-value. At that x the squared bracket is 0, so y is the number added on the end.
   y = 3
3. Say whether it is a maximum or minimum. A positive coefficient of the bracket gives a U-shape (minimum); a negative one gives an n-shape (maximum).
   (5, 3): minimum

Check. Expand and substitute x = 5: y = 5² − 10 × 5 + 28 = 25 − 50 + 28 = 3 ✓

Watch out. Opposite sign for x, same sign for y.

Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).

Method. The turning point of y = a(x + p)² + q is (−p, q). The bracket gives the x-value (opposite sign) and the number on the end gives the y-value.

1. Find the x-value. It makes the bracket equal to 0, so it has the opposite sign to the number inside the bracket.
   x = −2
2. Find the y-value. At that x the squared bracket is 0, so y is the number added on the end.
   y = −7
3. Say whether it is a maximum or minimum. A positive coefficient of the bracket gives a U-shape (minimum); a negative one gives an n-shape (maximum).
   (−2, −7): minimum

Check. Expand and substitute x = −2: y = (−2)² + 4 × (−2) − 3 = 4 − 8 − 3 = −7 ✓

Watch out. Opposite sign for x, same sign for y.

Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).

Method. The turning point of y = a(x + p)² + q is (−p, q). The bracket gives the x-value (opposite sign) and the number on the end gives the y-value.

1. Find the x-value. It makes the bracket equal to 0, so it has the opposite sign to the number inside the bracket.
   x = 1
2. Find the y-value. At that x the squared bracket is 0, so y is the number added on the end.
   y = 4
3. Say whether it is a maximum or minimum. A positive coefficient of the bracket gives a U-shape (minimum); a negative one gives an n-shape (maximum).
   (1, 4): maximum

Check. Expand and substitute x = 1: y = −(1)² + 2 × 1 + 3 = −1 + 2 + 3 = 4 ✓

Watch out. Opposite sign for x, same sign for y.