Section 4: Quadratics

For y = k(x + p)^2 + q the turning point is (-p, q): read the sign inside the bracket carefully.

Quick checks: multiple choice

Question 1Paper 1 style · no calculator

The graph shows a parabola with equation

y

= kx^2

. It passes through the point

(3, 18)

. Find the value of

k

Question 2Paper 1 style · no calculator

The parabola shown has a minimum turning point at

(2, -3)

. Its equation is of the form

y

= (x + p)^2 + q

. Find its equation.

Question 3Paper 1 style · no calculator

The parabola shown has a maximum turning point at

(1, 8)

and crosses the

x

axis at

(-1, 0)

and

(3, 0)

. Its equation is of the form

y

= k(x + p)^2 + q

. Find its equation.

Question 4Paper 2 style · calculator

The cross-section of a satellite dish is modelled by

y

= kx^2

. The curve passes through the point

(2.5, 0.75)

. Find the value of

k

Question 5Paper 1 style · no calculator

State the coordinates and the nature of the turning point of

y

= 5 - (x + 3)^2

Question 6Paper 1 style · no calculator

Find the coordinates of the points where the graph of

y

= (2x - 1)(x + 3)

meets the axes.

Question 7Paper 1 style · no calculator

Find the coordinates of the turning point of the graph of

y

= (x - 1)(x - 7)

Question 8Paper 1 style · no calculator

By writing

x^2 - 6x + 14

in the form

(x + p)^2 + q

, find the coordinates of the turning point of

y

= x^2 - 6x + 14

Question 9Paper 2 style · calculator

A ball is thrown from the top of a cliff. Its height above the beach,

h

metres,

t

seconds after it is thrown is

h

= 20.5 - 4.9(t - 1.4)^2

. Find, to

2

decimal places, the time when the ball lands on the beach.

Question 10Paper 1 style · no calculator

Find the coordinates of the point where the graph of

y

= 3(x + 1)^2 - 7

crosses the

y

axis.

Question 11Paper 1 style · no calculator

Solve

(3x + 2)(x - 5)

= 0

Question 12Paper 1 style · no calculator

Solve

2x^2 - 5x - 12

= 0

Question 13Paper 1 style · no calculator

Solve

x^2

= 4x + 21

Question 14Paper 1 style · no calculator

The graph of

y

= x^2 - 2x - 8

is shown. From the graph, solve

x^2 - 2x - 8

= 0

Question 15Paper 1 style · no calculator

A rectangle is

(x + 5)

cm long and

(x - 1)

cm wide. Its area is

40

^2

. Find the value of

x

Question 16Paper 2 style · calculator

A stone is thrown from a cliff top. Its height above the sea,

h

metres, after

t

seconds is

h

= 30 + 5t - 5t^2

. By factorising, find when the stone hits the sea.

Question 17Paper 2 style · calculator

Solve

3x^2 - 4x - 2

= 0

, giving your answers to

2

decimal places.

Question 18Paper 2 style · calculator

Solve

x(x + 4)

= 9

, giving your answers to

2

decimal places.

Question 19Paper 2 style · calculator

Solve

1.5x^2 + 2x - 4

= 0

, giving your answers to

1

decimal place.

Question 20Paper 1 style · no calculator

Solve

x^2 - 4x - 1

= 0

, giving your answers in simplest surd form.

Question 21Paper 2 style · calculator

A rectangular garden is

3

m longer than it is wide. Its area is

50

^2

. Find its width, to

2

decimal places.

Question 22Paper 1 style · no calculator

Determine the nature of the roots of

x^2 - 5x + 7

= 0

Question 23Paper 1 style · no calculator

The equation

2x^2 + 8x + k

= 0

has one repeated real root. Find the value of

k

Question 24Paper 2 style · calculator

Find the discriminant of

0.5x^2 - 2.4x + 2.88

= 0

and state the nature of its roots.

Question 25Paper 1 style · no calculator

Which of these equations has two real and distinct roots?

Paper-style questions: Paper 1, then Paper 2

Features of a parabolaPaper 1 style, no calculator · 8 marks

A parabola has equation

y

= x^2 - 8x + 12

(a) Write

x^2 - 8x + 12

in the form

(x + p)^2 + q

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) State the coordinates and the nature of the turning point of the parabola. [2]

x^2 - 8x + 12

= 0

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) State the coordinates of the point where the parabola crosses the

y

axis. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) State the equation of the axis of symmetry of the parabola. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

A rectangular lawnPaper 2 style, calculator · 9 marks

A rectangular lawn is

(x + 4)

metres long and

(2x - 1)

metres wide. Its area is

30

square metres.

(a) Write an equation for the area in the form

ax^2 + bx + c

= 0

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Calculate the discriminant of this equation. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Solve the equation, giving your answers to

2

decimal places. [3]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(e) Find the length of the lawn, to

2

decimal places. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).