AI generatedAlgebraGCSE Higher

Finding a quadratic minimum or maximum

Interpret the turning point and the leading coefficient.

  • Roots are where . The turning point is the minimum or maximum.
  • The axis of symmetry is the vertical line through the turning point.
  • A positive coefficient opens upwards; a negative coefficient opens downwards.

Example 1

Find the minimum or maximum value and the -value where it occurs.

Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the minimum.
Minimum value at .
Coordinate graph−8−6−4−22−8−6−4−2240

Example 2

Find the minimum or maximum value and the -value where it occurs.

Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the maximum.
Maximum value at .
Coordinate graph−6−4−22−15−12.5−10−7.5−5−2.50

Your turn

Question 1

Find the minimum or maximum value and the -value where it occurs.

Check answer
Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the minimum.
Minimum value at .
Coordinate graph−6−4−224−8−6−4−2240
Question 2

Find the minimum or maximum value and the -value where it occurs.

Check answer
Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the maximum.
Maximum value at .
Coordinate graph−4−224−15−12.5−10−7.5−5−2.50
Question 3

Find the minimum or maximum value and the -value where it occurs.

Check answer
Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the minimum.
Minimum value at .
Coordinate graph−4−2246−8−6−4−2240
Question 4

Find the minimum or maximum value and the -value where it occurs.

Check answer
Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the maximum.
Maximum value at .
Coordinate graph−2246−15−12.5−10−7.5−5−2.50
Question 5

Find the minimum or maximum value and the -value where it occurs.

Check answer
Complete the square to identify the turning point.The squared term is at least zero.The squared term is zero here, giving the minimum.
Minimum value at .
Coordinate graph−22468−8−6−4−2240
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