Robinson Tuition
Grade 9 · Higher · free GCSE maths worksheet

Turning points by completing the square

Sketching and finding the minimum.

12 original questions, 42 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

y = x2 + 6x + 4

By completing the square, find the coordinates of the turning point and say whether it is a maximum or a minimum.

3 marks
Show answer
(-3, -5) minimum
2

y = 3x2 − 12x + 5

Write y in the form a(x + b)2 + c and write down the coordinates of the turning point.

3 marks
Show answer
3(x - 2)2 - 7; (2, -7)
3

The curve C has equation y = (x − 3)2 + 5

Write down the coordinates of the turning point of C.

1 mark
Show answer
(3, 5)
4

  1. Write x2 − 6x + 11 in the form (x − a)2 + b, where a and b are integers.

  2. Hence find the coordinates of the turning point of the curve y = x2 − 6x + 11

3 marks
Show answer
a) (x - 3)2 + 2 b) (3, 2)
5

By completing the square, find the coordinates of the minimum point of the curve

y = x2 + 10x − 2

3 marks
Show answer
(-5, -27)
6

The curve y = x2 + bx + c passes through the points (0, 7) and (1, 4).

Find the coordinates of the turning point of the curve.

4 marks
Show answer
(2, 3)
7

The curve C has equation y = x2 − 4x − 5

  1. Write x2 − 4x − 5 in the form (x − a)2 + b

  2. Sketch C, showing clearly the coordinates of the turning point and the coordinates of the points where C crosses the axes.

−3−2−11234567−12−9−6−336912Oxy
5 marks
Show answer
a) (x - 2)2 - 9 b) U-shaped parabola, turning point (2, -9), crossing the x-axis at (-1, 0) and (5, 0) and the y-axis at (0, -5)
8

  1. Show that x2 + 5x + 9 can be written as (x + 52)2 + 114

  2. Write down the minimum value of x2 + 5x + 9

  3. Hence explain why the equation x2 + 5x + 9 = 0 has no real solutions.

5 marks
Show answer
a) (x + 5/2)2 = x2 + 5x + 25/4, so x2 + 5x + 9 = (x + 5/2)2 + 11/4 b) 11/4 c) Since (x + 5/2)2 >= 0, the expression is always >= 11/4, which is positive, so it is never 0
9

f(x) = x2 − 4x + 7

  1. Write f(x) in the form (x − a)2 + b and hence write down the coordinates of the turning point of y = f(x)
  2. Write down the coordinates of the turning point of y = f(x − 3)
  3. Find the equation of y = f(−x), giving your answer in the form y = x2 + bx + c

4 marks
Show answer
a) f(x) = (x - 2)2 + 3; turning point (2, 3) b) (5, 3) c) y = x2 + 4x + 7
10

y = −x2 + 6x − 4

By completing the square, find the coordinates of the turning point. State whether it is a maximum or a minimum.

3 marks
Show answer
(3, 5) maximum
11

A ball is thrown upwards. Its height, h metres, t seconds after it is thrown is given by

h = 1.5 + 20t − 5t2

By completing the square, work out the maximum height of the ball and the time at which it reaches this height.

4 marks
Show answer
21.5 m at t = 2 s
12

The graph of y = x2 + bx + c has a turning point at (3, −2).

Find the value of b and the value of c.

4 marks
Show answer
b) -6 c) 7

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

All Grade 9 topics