GCSE Maths · Higher · Chapter 23: Graphs

Distance-time and velocity-time graphs, areas under curves, circles, other graphs and transformations

Grades 4–9About 90 minutesCalculator allowed96 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Work out the gradient of the straight line through $(1, 3)$ and $(5, 11)$ .
[2 marks]
Answer

A2
A trapezium has parallel sides of 6 cm and 10 cm. The distance between them is 4 cm. Work out its area.
[2 marks]
Answer
A3
A train travels 252 km in 3.6 hours. Work out its average speed.
[1 mark]
Answer
A4
$f (x) = x^{2} - 3$ . Work out $f (4)$ .
[1 mark]
Answer
A5
Write down the gradient of a line perpendicular to $y = 2 x + 1$ .
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

A car starts from rest and speeds up steadily to 12 m/s in 4 seconds. It travels at 12 m/s for 6 seconds, then slows down steadily and stops 5 seconds later. Sketch the velocity-time graph and work out the total distance travelled.

Hint The distance is the area under a velocity-time graph. Split it into two triangles and a rectangle.

B1
A cyclist starts from rest and speeds up steadily to 8 m/s in 5 seconds. She travels at 8 m/s for 10 seconds, then slows down steadily and stops 3 seconds later.
Work out her acceleration in the first 5 seconds, and the total distance she travels.
[4 marks]
Answer $\dots\dots$ m/s² $\dots\dots$ m

Example 2

Use 4 strips of equal width to estimate the area under the curve $y = x^{2} + 1$ between $x = 0$ and $x = 4$ .

Hint Work out

y

x = 0, 1, 2, 3, 4

. Each strip is a trapezium of width 1.

B2
Use 3 strips of equal width to estimate the area under the curve $y = 10 - x^{2}$ between $x = 0$ and $x = 3$ .
[3 marks]
Answer

Example 3

Find the equation of the tangent to the circle $x^{2} + y^{2} = 25$ at the point $(3, 4)$ .

Hint The tangent is perpendicular to the radius. Find the gradient of the radius first.

B3
Find the equation of the tangent to the circle $x^{2} + y^{2} = 52$ at the point $(4, 6)$ . Give your answer in the form $a x + b y = c$ .
[3 marks]
Answer

Example 4

The graph of $y = f (x)$ has a turning point at $(2, - 5)$ . Write down the turning point of $y = f (x + 3)$ and of $y = - f (x)$ .

Hint

f (x + 3)

moves the graph 3 to the left.

- f (x)

reflects it in the

x

-axis.

B4
The graph of $y = f (x)$ has a turning point at $(- 4, 3)$ . Write down the turning point of $y = f (x) - 6$ and of $y = f (- x)$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$ $(\dots\dots, \dots\dots)$

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Leah cycles from home to a park, rests, and then cycles home. The graph shows her journey.
(a)
Work out her speed on the way to the park, in km/h.
[2 marks]
Answer
(b)
For how many minutes does she rest?
[1 mark]
Answer
(c)
Work out her speed on the way home, in km/h.
[2 marks]
Answer
Total for C1: 5 marks

C2
Here are four sketch graphs. Match each equation to its graph.
Not drawn accurately
(a)
$y = x^{3} - 4 x$
[1 mark]
Answer
(b)
$y = 4 - x^{2}$
[1 mark]
Answer
(c)
$y = \frac{3}{x}$
[1 mark]
Answer
(d)
$y = 2^{x}$
[1 mark]
Answer
Total for C2: 4 marks
C3
A circle has equation $x^{2} + y^{2} = 72$ .
(a)
Write down its radius in the form $a 2$ .
[2 marks]
Answer
(b)
Is the point $(5, 7)$ inside the circle, on it, or outside it? Give a reason.
[2 marks]
Answer
Total for C3: 4 marks
C4
The sketch shows the curve $y = a b^{x}$ , where $b > 0$ . The curve passes through $(0, 5)$ and $(2, 45)$ .
Find the value of $a$ and the value of $b$ .
[3 marks]
Not drawn accurately
Answer $a = \dots\dots$ $b = \dots\dots$
C5
Describe fully the single transformation that maps the graph of $y = x^{2}$ onto the graph of $y = (x - 3)^{2} + 2$ .
[2 marks]
Answer
C6
Sketch the graph of $y = (x - 1) (x + 2) (x - 3)$ . Show the coordinates of the points where it crosses the axes.
[3 marks]
C7
The table gives the velocity of a car at times during the first 6 seconds of a journey. Its velocity-time graph is a smooth curve whose gradient keeps falling.
(a)
Use 3 strips of equal width to estimate the distance the car travels in the first 6 seconds.
[3 marks]
Answer
(b)
Is your answer an overestimate or an underestimate? Give a reason.
[1 mark]
Answer
Total for C7: 4 marks
C8
A circle has its centre at the origin and passes through the point $(- 5, 12)$ . Write down its equation.
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Kai says, "The graph of $y = f (x + 4)$ is the graph of $y = f (x)$ moved 4 units to the right."
Kai is wrong. Explain why, and describe the transformation fully.
[2 marks]
Answer

D2
The diagram shows the circle $x^{2} + y^{2} = 20$ and the point $P (2, 4)$ on it.
Not drawn accurately
(a)
Find the equation of the tangent to the circle at $P$ . Give your answer in the form $a x + b y = c$ .
[3 marks]
Answer
(b)
The tangent meets the $x$ -axis at $A$ and the $y$ -axis at $B$ . Work out the area of triangle $O A B$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
A train starts from rest and speeds up steadily to 20 m/s in $T$ seconds. It travels at 20 m/s for 60 seconds, then slows down steadily and stops 25 seconds later. The train travels 1800 m altogether.
Work out the value of $T$ . You must show your working.
[4 marks]
Not drawn accurately
Answer
D4
The graph shows the velocity, $v$ m/s, of a model rocket car $t$ seconds after it starts, where $v = 12 t - t^{2}$ for $0 \leq t \leq 10$ .
(a)
Use 5 strips of equal width to estimate the distance the car travels in the first 10 seconds.
[3 marks]
Answer
(b)
Is your answer to part (a) an overestimate or an underestimate? Give a reason.
[1 mark]
Answer
(c)
By drawing a tangent, estimate the acceleration of the car when $t = 4$ .
[2 marks]
Answer
Total for D4: 6 marks
D5
The value of a van, £ $V$ , is modelled by $V = 18000 \times 0.8 5^{t}$ , where $t$ is its age in years.
(a)
Work out the value of the van when it is 3 years old.
[2 marks]
Answer
(b)
After how many complete years is the value first less than £8000?
[2 marks]
Answer
Total for D5: 4 marks
D6
Solve the simultaneous equations
$y = 2 x x^{2} + y^{2} = 45$
You must show your working.
[4 marks]
Answer $(\dots\dots, \dots\dots)$ $(\dots\dots, \dots\dots)$
D7
The graph of $y = x^{2}$ is reflected in the $x$ -axis. The new graph is then translated by $(34)$ .
Show that the final graph has equation $y = - x^{2} + 6 x - 5$ .
[3 marks]
D8
The curve $y = \frac{k}{x}$ passes through the point $(4, - 3)$ .
(a)
Find the value of $k$ .
[1 mark]
Answer
(b)
Explain why the curve never meets the line $y = x$ .
[2 marks]
Answer
Total for D8: 3 marks
D9
A toy car moves in a straight line. The distance, $d$ metres, it has travelled after $t$ seconds is $d = 2 t^{2}$ .
(a)
Work out the average speed of the car between $t = 1$ and $t = 4$ .
[2 marks]
Answer
(b)
Estimate the speed of the car at $t = 3$ by working out its average speed between $t = 2.9$ and $t = 3.1$ .
[2 marks]
Answer
Total for D9: 4 marks
D10
The sketch shows the graph of $y = f (x)$ . It crosses the $x$ -axis at $x = - 2$ and $x = 3$ , and the $y$ -axis at $y = 6$ .
Not drawn accurately
(a)
Write down the coordinates of the points where the graph of $y = f (x - 1)$ crosses the $x$ -axis.
[2 marks]
Answer $(\dots\dots, \dots\dots)$ $(\dots\dots, \dots\dots)$
(b)
Write down the coordinates of the point where the graph of $y = - f (x)$ crosses the $y$ -axis.
[1 mark]
Answer $(\dots\dots, \dots\dots)$
Total for D10: 3 marks

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Two tangents to the circle $x^{2} + y^{2} = 25$ have gradient $\frac{3}{4}$ . Find their equations.
[3 marks]
Hint 1 · What to try
A tangent is perpendicular to the radius, so find where the radius has gradient $- \frac{4}{3}$ .
Hint 2 · The first line
The radius has gradient $- \frac{4}{3}$ at $(- 3, 4)$ and at $(3, - 4)$ .
Hint 3 · The full method
The tangent at $(- 3, 4)$ is $y - 4 = \frac{3}{4} (x + 3)$ , so $y = \frac{3}{4} x + \frac{25}{4}$ . The tangent at $(3, - 4)$ is $y = \frac{3}{4} x - \frac{25}{4}$ .
Answer

E2
The graph of $y = 2^{x}$ is translated by $(p q)$ . The new graph passes through $(2, 5)$ and $(4, 11)$ .
Find the value of $p$ and the value of $q$ .
[3 marks]
Hint 1 · What to try
Write the equation of the new graph using $p$ and $q$ .
Hint 2 · The first line
The new graph is $y = 2^{x - p} + q$ , so $2^{2 - p} + q = 5$ and $2^{4 - p} + q = 11$ .
Hint 3 · The full method
Subtracting, $2^{2 - p} (4 - 1) = 6$ , so $2^{2 - p} = 2$ . Then $p = 1$ and $q = 3$ .
Answer $p = \dots\dots$ $q = \dots\dots$
E3
$P$ is a point on the circle $x^{2} + y^{2} = 50$ . The tangent to the circle at $P$ passes through the point $(10, 0)$ .
Find the two possible positions of $P$ .
[3 marks]
Hint 1 · What to try
Call $P$ the point $(a, b)$ and write down the equation of the tangent at $P$ .
Hint 2 · The first line
The tangent at $(a, b)$ is $a x + b y = 50$ . It passes through $(10, 0)$ , so $10 a = 50$ .
Hint 3 · The full method
$a = 5$ , and $25 + b^{2} = 50$ gives $b = 5$ or $b = - 5$ . So $P$ is $(5, 5)$ or $(5, - 5)$ .
Answer $(\dots\dots, \dots\dots)$ $(\dots\dots, \dots\dots)$
E4
Find where the curve $y = \frac{1}{x - 2} + 3$ crosses each axis, and write down the equations of its two asymptotes.
[3 marks]
Hint 1 · What to try
The curve is $y = \frac{1}{x}$ translated. Find the translation first.
Hint 2 · The first line
It is the translation by $(23)$ , so the asymptotes are $x = 2$ and $y = 3$ .
Hint 3 · The full method
At $x = 0$ , $y = - \frac{1}{2} + 3 = \frac{5}{2}$ . At $y = 0$ , $\frac{1}{x - 2} = - 3$ , so $x = \frac{5}{3}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out a speed from a distance-time graph, and an acceleration and a distance from a velocity-time graph.
I can estimate the area under a curve with trapezia, say what it means, and say whether the estimate is too big or too small.
I can use the equation $x^{2} + y^{2} = r^{2}$ and find the tangent to a circle at a given point.
I can recognise cubic, reciprocal and exponential graphs from their equations.
I can translate and reflect the graph of $y = f (x)$ and say where its points move to.

Answers and mark scheme

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