CHAPTER 3 QUADRATIC FUNCTIONS FORM 4

Paper 1

1. The quadratic function f(x) = a(x+p)2 + q, where a, p and q are constants, has a maximum value of 5. The equation of the axis of symmetry is x=3.

State (a) the range of values of a, (b) the value of p (c) the value of q [3 marks]

2. Find the range of values of x for which (x 4)2 < 12 3x [3 marks]

3. Find the range of values of x for which 2x2 3 5x. [3 marks]

4. The quadratic function f(x) = x2 6x + 5 can be expressed in the form f(x) = (x + m)2 + n where m and n are constants. Find the value of m and of n. [3 marks] 5. The following diagram shows the graph of quadratic function y = g(x). The straight line y = 9 is a tangent to the curve y = g(x).

(a) Write the equation of the axis of symmetry of the curve

(b) Express g(x) in the form (x + b)2 + c where b and c are constant. [3 marks]

6. The following diagram shows the graph of a quadratic function , where p is a constant.

9

x

y

O 15

y = 9

A(1, q).

0 x

y

CHAPTER 3 QUADRATIC FUNCTIONS FORM 4

The curve has a maximum point at A(1, q), where q is a constant. State

(a) the value of p, (b) the value of q, (c) the equation of the axis of symmetry.

[3 marks ]

7. Find the range of values of x for which x(2 x) 15. [3 marks ]

8. The quadratic equation x(p x ) = x + 4 has no real roots. Find the range of values of p [3 marks ]

Paper 2

1. Diagram below shows the curve of a quadratic function .

A is the point of intersection of the quadratic graph and y-axis. The x-intercepts are 6 and 2.

(a) State the value of r and of p. [2 marks]

(b) The function can be expressed in the form , find the value of q and of k.

[4 marks]

(c) Determine the range of values of x if f(x)< 6. [2 marks]

10

x

y

O 26

A(0, r)

(p, q)

CHAPTER 3 QUADRATIC FUNCTIONS FORM 4

Answers ( Paper 1)

Q Solution Marks1 (a) a<0 1 (b) p=3 1 (c) q=5 12 x2 5x + 4 < 0 1

(x 1)(x 4) < 0 1 1<x<4 1

3 2x2 + 5x. 3 0 1(2x 1)(x + 3) 0 1 3 x ½ 1

4 f(x) = x2 6x + 32 32 + 5 = (x 3)2 4

1

m = 3 n = 4 1,15 (a) x = 2 1 (b)

g(x) = (x + 2)2 91,1

6 (a) p = 1 1 (b) q =5 1 (c) x=1 17 x2 2x 15 0 1

(x + 3)(x 5) 0 1 x3, x5 1

8 x2 + (1 p)x + 4 =0(1 p)2 4(1)(4) < 0 1 p2 2p 15 < 0

(p+3)(p 5) <0 13 < p < 5 1

Answer(Paper 2)1 (a) r = 6, 1

p = 2 1 (b) (Expand or completing the square) 1

or k=p

1q = 8 1k =2 1

(c) x2 + 4x<0

x(x + 4)<0

1

4 < x < 0 1

11

4 0 x