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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Quadratic functions

Completing the square, the discriminant, sketching, maximum and minimum values.

12 original questions, 51 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

Write x2 − 8x + 3 in the form (x + p)2 + q, where p and q are integers.

2 marks
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(x - 4)2 - 13
2

Write 2x2 + 12x + 7 in the form a(x + b)2 + c, where a, b and c are integers.

3 marks
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2(x + 3)2 - 11
3

f(x) = 3x2 − 5x + 4

Find the value of the discriminant of f(x) and hence state the number of real roots of the equation f(x) = 0

2 marks
Show answer
-23, so no real roots
4

f(x) = x2 + 6x + 14

  1. Write f(x) in the form (x + a)2 + b, where a and b are integers.
  2. Sketch the curve with equation y = f(x), showing the coordinates of the turning point and of any point where the curve meets the coordinate axes.
xy
5 marks
Show answer
a) (x + 3)2 + 5 b) U-shaped curve, minimum (-3, 5), y-intercept (0, 14), no x-intercepts
5

The equation 3x2 − 6x + k = 0, where k is a constant, has two distinct real roots.

Find the range of possible values of k.

3 marks
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k < 3
6

Prove that 4x2 − 12x + 11 > 0 for all real values of x.

3 marks
Show answer
4x2 - 12x + 11 = 4(x - 32)2 + 2, which is at least 2, so it is always positive
7

g(x) = 5 + 8x − 2x2

  1. Write g(x) in the form a − b(x − c)2, where a, b and c are integers.
  2. Hence state the maximum value of g(x) and the value of x at which it occurs.
5 marks
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a) 13 - 2(x - 2)2 b) maximum value 13, when x = 2
8

The figure shows a sketch of the curve with equation y = ax2 + bx + c, where a, b and c are constants.

The curve has a minimum turning point at (3, −8) and crosses the x-axis at (1, 0).

  1. Find the values of a, b and c.
  2. Find the coordinates of the other point where the curve crosses the x-axis.
xy(1, 0)(3, -8)
4 marks
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a) a = 2, b = -12, c = 10 b) (5, 0)
9

The equation x2 + 2px + (3p + 4) = 0, where p is a constant, has equal roots.

  1. Find the possible values of p.
  2. For each value of p, find the root of the equation.
5 marks
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a) p = 4 or p = -1 b) p = 4 gives x = -4; p = -1 gives x = 1
10

Amara kicks a ball from the top of a wall. The height of the ball above the ground, h metres, t seconds after it is kicked is modelled by

h = 2 + 12t − 5t2

  1. Write h in the form A − B(t − C)2, where A, B and C are constants.
  2. Hence find, according to the model, the maximum height of the ball and the time at which it reaches it.
  3. Find the time when the ball hits the ground, giving your answer to 3 significant figures.
  4. Give one limitation of the model.
8 marks
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a) h = 9.2 - 5(t - 1.2)2 b) 9.2 m, at t = 1.2 seconds c) t = 2.56 seconds d) e.g. it ignores air resistance, or the ground may not be level
11
  1. Write x2 − 6x + 13 in the form (x + p)2 + q, where p and q are integers.
  2. Hence find the maximum value of 10x2 − 6x + 13, the value of x at which it occurs, and the range of possible values of 10x2 − 6x + 13 for real x.
5 marks
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a) (x - 3)2 + 4 b) maximum 52 when x = 3; 0 < 10x2 - 6x + 13 ≤ 52
12

f(x) = x2 − 2kx + 3k + 4, where k is a constant.

  1. Write f(x) in the form (x + a)2 + b, where a and b are given in terms of k.
  2. Given that the curve y = f(x) lies entirely above the x-axis, find the range of possible values of k.
  3. When k = 4, write down the coordinates of the turning point of y = f(x) and describe how the curve meets the x-axis.
6 marks
Show answer
a) (x - k)2 - k2 + 3k + 4 b) -1 < k < 4 c) (4, 0); the curve touches the x-axis at this point

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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