Sec 3 E-Math Graphs of functions worksheet with answers

A free Sec 3 E-Math worksheet on graphs of functions: 7 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 3 E-Math worksheet: Graphs of functions

Total: 30 marksSuggested time: 45 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

The diagram shows the speed–time graph of a car that starts from rest. The maximum speed,

V

, of the car is 108 km/h. Find the total distance, in metres, travelled by the car.

Answerm[2]

The diagram shows the graph of a function. Which of these could be its equation?

Answer[2]

The diagram shows the graph of

y = -x^{3} + 3x^{2} - 1

. Find the range of values of

k

for which the equation

-x^{3} + 3x^{2} - 1 = k

has three solutions.

Answer[2]

Each diagram shows the curve

y = -x^{2} + 4

, the point

P

on it where

x = 1

, and a straight line. In which diagram is the line the tangent to the curve at

P

Answer[3]

Section B

The diagram shows part of the graph of

y = -x^{3} + 3x + 1

, with the tangent to the curve at

x = 0

drawn.

(a)Find the value of

y

when

x = -2.5

, correct to 1 decimal place.

Answer (a)[1]

(b)Use the tangent to find the gradient of the curve at

x = 0

Answer (b)[2]

(c)By drawing a suitable straight line on the graph, the equation

-x^{3} + 4x = 0

can be solved. Find the equation of this straight line.

Answer (c)[2]

A motorcycle starts from rest and accelerates uniformly to a speed of

15

m/s in

4

seconds. It then travels at this speed. As the motorcycle starts, a lorry travelling at a constant speed of

12

m/s passes it in the same direction. The diagram shows the speed–time graphs of the two vehicles.

(a)Find the acceleration of the motorcycle in the first

4

seconds.

Answer (a)m/s²[1]

(b)Find the time at which the motorcycle and the lorry have the same speed.

Answer (b)s[1]

(c)Show that the motorcycle catches up with the lorry after

10

seconds.

[2]

(d)Find how far from the starting point the motorcycle catches up with the lorry.

Answer (d)m[1]

(e)Before the motorcycle catches up, the lorry is ahead. Find the greatest distance between them.

Answer (e)m[2]

The table shows some values of

x

and the corresponding values of

y

for

y = -x^{2} + 4x

$x$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$
$y$	$-5$	$0$	$p$	$4$	$3$	$0$	$-5$

(a)Find the value of

p

Answer (a)[1]

(b)The grid above shows the graph of

y = -x^{2} + 4x

for

-1 \leq x \leq 5

. Use the graph to solve the equation

-x^{2} + 4x = -4

, giving your answers correct to 1 decimal place.

Answer (b)[2]

(c)By drawing a tangent, find the gradient of the curve at the point where

x = 1

Answer (c)[2]

(d)By drawing a suitable straight line on the grid, the equation

x^{2} - 2x - 1 = 0

can be solved. Find the equation of this straight line.

Answer (d)[2]

(e)Hence solve the equation

x^{2} - 2x - 1 = 0

, giving your answers correct to 1 decimal place.

Answer (e)[2]

END OF PAPER

Answers and worked solutions

Question 1

Answer:

\phantom{\text{Distance}} = 1125 \text{ m}

[2]

V = \dfrac{108 \times 1000}{3600}

\phantom{V} = 30 \text{ m/s}

\text{Distance} = \dfrac{1}{2} \times (25 + 50) \times 30

\phantom{\text{Distance}} = 1125 \text{ m}

Question 2

Answer: D[2]

y = 2^{x}

is exponential.At

x = 0

y = 2^{0}

y = 1

: it crosses the

y

-axis at

(0, 1)

and stays above the

x

-axis.

y = 2^{x}

, option

D

Question 3

Answer:

-1 < k < 3

[2]

Maximum point:

y = 3

Minimum point:

y = -1

-1 < k < 3

Question 4

Answer: A[3]

Rule out any line that misses

P

.Near

P

, the curve stays on one side of the line.Diagram

A

: the line is the tangent at

P

.Diagram

B

: the line cuts the curve at two points: it is a chord, not a tangent.Diagram

C

: the line goes through

P

but crosses the curve there, at the wrong angle.Diagram

D

: the line touches the curve, but not at

P

.Diagram

A

Question 5

(a)Answer:

9.1

[1]

y = 9.125

= 9.1

(1 d.p.)

(b)Answer:

3

[2]

The tangent passes through

(-1, -2)

and

(1, 4)

.Gradient

= \dfrac{6}{2}

= 3

(c)Answer:

y = -x + 1

[2]

-x^{3} + 4x = 0

-x^{3} + 3x + 1 = -x + 1

Draw

y = -x + 1

Question 6

(a)Answer:

3.75

m/s²[1]

\text{Acceleration} = \dfrac{15}{4}

\phantom{\text{Acceleration}} = 3.75

m/s²

(b)Answer:

3.2

s[1]

3.75t = 12

t = 3.2

(c)[2]

After

T

seconds, distance of the motorcycle

= \dfrac{1}{2} \times 4 \times 15 + 15(T - 4)

Distance of the lorry

= 12T

30 + 15T - 60 = 12T

3T = 30

T = 10

(shown)

(d)Answer:

120

m[1]

\text{Distance} = 12 \times 10

\phantom{\text{Distance}} = 120

(e)Answer:

19.2

m[2]

The lorry pulls away while it is faster; the gap is greatest when the speeds are equal, at

t = 3.2

.Lorry:

12 \times 3.2 = 38.4

m; motorcycle:

\dfrac{1}{2} \times 3.2 \times 12 = 19.2

\text{Gap} = 38.4 - 19.2

\phantom{\text{Gap}} = 19.2

Question 7

(a)Answer:

p = 3

[1]

p = -(1)^{2} + 4(1)

\phantom{p} = 3

(b)Answer:

x = -0.8

x = 4.8

[2]

Draw the line

y = -4

on the grid.It meets the curve where

x = -0.8

x = 4.8

(c)Answer:

2

(accept

1.8

2.2

)[2]

Draw the tangent at

(1, 3)

. It passes through

(0, 1)

and

(2, 5)

.Gradient

= \dfrac{5 - 1}{2 - 0}

= 2

(d)Answer:

y = 2x - 1

[2]

x^{2} - 2x - 1 = 0

-x^{2} + 2x + 1 = 0

(multiply by

-1

)

-x^{2} + 4x = 2x - 1

Draw the line

y = 2x - 1

(e)Answer:

x = -0.4

x = 2.4

[2]

Draw

y = 2x - 1

through

(0, -1)

and

(2, 3)

.It meets the curve where

x = -0.4

x = 2.4

Practice questions by skill

Sec 3 E-Math Graphs of functions worksheet with answers

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Graphs of functions notes. Or practise it online with step-by-step help when you get one wrong.

Section B

The diagram shows part of the graph of

y = -x^{3} + 3x + 1

, with the tangent to the curve at

x = 0

drawn.

(a)Find the value of

y

when

x = -2.5

, correct to 1 decimal place.

Answer (a)[1]

(b)Use the tangent to find the gradient of the curve at

x = 0

Answer (b)[2]

(c)By drawing a suitable straight line on the graph, the equation

-x^{3} + 4x = 0

can be solved. Find the equation of this straight line.

Answer (c)[2]

x

-1

0

1

2

3

4

5

y

-5

0

p

4

3

0

-5

Answers and worked solutions

Question 1

Answer:

\phantom{\text{Distance}} = 1125 \text{ m}

[2]

V = \dfrac{108 \times 1000}{3600}

\phantom{V} = 30 \text{ m/s}

\text{Distance} = \dfrac{1}{2} \times (25 + 50) \times 30

\phantom{\text{Distance}} = 1125 \text{ m}

Question 2

Answer: D[2]

y = 2^{x}

is exponential.At

x = 0

y = 2^{0}

y = 1

: it crosses the

y

-axis at

(0, 1)

and stays above the

x

-axis.

y = 2^{x}

, option

D

Question 3

Answer:

-1 < k < 3

[2]

Maximum point:

y = 3

Minimum point:

y = -1

-1 < k < 3

Question 4

Answer: A[3]

Rule out any line that misses

P

.Near

P

, the curve stays on one side of the line.Diagram

A

: the line is the tangent at

P

.Diagram

B

: the line cuts the curve at two points: it is a chord, not a tangent.Diagram

C

: the line goes through

P

but crosses the curve there, at the wrong angle.Diagram

D

: the line touches the curve, but not at

P

.Diagram

A

Question 5

(a)Answer:

9.1

[1]

y = 9.125

= 9.1

(1 d.p.)

(b)Answer:

3

[2]

The tangent passes through

(-1, -2)

and

(1, 4)

.Gradient

= \dfrac{6}{2}

= 3

(c)Answer:

y = -x + 1

[2]

-x^{3} + 4x = 0

-x^{3} + 3x + 1 = -x + 1

Draw

y = -x + 1

Question 6

(a)Answer:

3.75

m/s²[1]

\text{Acceleration} = \dfrac{15}{4}

\phantom{\text{Acceleration}} = 3.75

m/s²

(b)Answer:

3.2

s[1]

3.75t = 12

t = 3.2

(c)[2]

After

T

seconds, distance of the motorcycle

= \dfrac{1}{2} \times 4 \times 15 + 15(T - 4)

Distance of the lorry

= 12T

30 + 15T - 60 = 12T

3T = 30

T = 10

(shown)

(d)Answer:

120

m[1]

\text{Distance} = 12 \times 10

\phantom{\text{Distance}} = 120

(e)Answer:

19.2

m[2]

The lorry pulls away while it is faster; the gap is greatest when the speeds are equal, at

t = 3.2

.Lorry:

12 \times 3.2 = 38.4

m; motorcycle:

\dfrac{1}{2} \times 3.2 \times 12 = 19.2

\text{Gap} = 38.4 - 19.2

\phantom{\text{Gap}} = 19.2

Question 7

(a)Answer:

p = 3

[1]

p = -(1)^{2} + 4(1)

\phantom{p} = 3

(b)Answer:

x = -0.8

x = 4.8

[2]

Draw the line

y = -4

on the grid.It meets the curve where

x = -0.8

x = 4.8

(c)Answer:

2

(accept

1.8

2.2

)[2]

Draw the tangent at

(1, 3)

. It passes through

(0, 1)

and

(2, 5)

.Gradient

= \dfrac{5 - 1}{2 - 0}

= 2

(d)Answer:

y = 2x - 1

[2]

x^{2} - 2x - 1 = 0

-x^{2} + 2x + 1 = 0

(multiply by

-1

)

-x^{2} + 4x = 2x - 1

Draw the line

y = 2x - 1

(e)Answer:

x = -0.4

x = 2.4

[2]

Draw

y = 2x - 1

through

(0, -1)

and

(2, 3)

.It meets the curve where

x = -0.4

x = 2.4