GCSE Maths · Higher · Chapter 23: Graphs · Lesson 5

Transformations of y = f(x)

Grades 6–9About 50 minutesNo calculator32 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Rotate $(3, - 1)$ through $18 0^{\circ}$ about the origin.
[1 mark]
Answer
A2
From chapter 17
Solve $(x - 3) (x - 8) \leq 0$ .
[1 mark]
Answer
A3
From chapter 23
Write down the radius of the circle $x^{2} + y^{2} = 25$ .
[1 mark]
Answer
A4
From chapter 15
Solve $8 x - 4 = 6 x + 4$ .
[1 mark]
Answer

Example, then your turn

Grades 6–7

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

Example 1

The point $(2, 5)$ lies on the graph of $y = f (x)$ . Write down its image on the graph of $y = f (x - 4)$ , on $y = - f (x)$ and on $y = f (x) + 3$ .

Hint Inside the bracket moves

x

, the opposite way to its sign. Outside moves

y

B1
The point $(- 3, 4)$ lies on the graph of $y = f (x)$ . Write down its image on the graph of $y = f (x + 5)$ , on $y = f (- x)$ and on $y = f (x) - 6$ .
[3 marks]
Answer $(\dots, \dots)$ $(\dots, \dots)$ $(\dots, \dots)$

Example 2

The graph of $y = x^{2}$ is translated by $(- 3 2)$ . Write down the equation of the new graph.

Hint Moving left 3 puts

+ 3

inside the bracket. Moving up 2 adds 2 outside.

B2
The graph of $y = x^{3}$ is translated by $(2 - 5)$ . Write down the equation of the new graph.
[2 marks]
Answer

Practice

Grades 6–7

C1
The sketch shows $y = f (x)$ . Write down the turning point of $y = f (x - 5)$ .
[1 mark]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$
C2
The sketch shows $y = sin x$ , with its maximum point $(90, 1)$ . Write down the maximum point of $y = sin x + 3$ and of $y = sin (x - 40)$ .
[2 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$

C3
The graph of $y = x^{2} + 5 x$ is reflected in the $y$ -axis. Write down the equation of the new graph.
[1 mark]
Answer
C4
The graph of $y = x^{3}$ is translated by $(0 - 8)$ . Where does the new graph cross the $x$ -axis?
[2 marks]
Answer
C5
The sketch shows $y = f (x)$ and a curve $G$ of the same shape. Write down the equation of $G$ in terms of $f$ .
[1 mark]
Answer
C6
The sketch shows $y = f (x)$ . Write down the points where the graph of $y = f (- x)$ crosses the axes.
[2 marks]
Not drawn accurately
Answer $(\dots, \dots)$ , $(\dots, \dots)$ , $(\dots, \dots)$

Exam-style questions

Grades 7–9

D1
Mia says, "The graph of $y = f (x) - 3$ is the graph of $y = f (x)$ moved 3 units to the left."
Explain why Mia is wrong, and describe the transformation fully.
[2 marks]

D2
A curve has equation $y = x^{2} - 14 x + 40$ .
(a)
Write $x^{2} - 14 x + 40$ in the form $(x - a)^{2} - b$ , where $a$ and $b$ are positive integers.
[2 marks]
Answer
(b)
Hence describe fully the single transformation that maps the graph of $y = x^{2}$ onto the graph of $y = x^{2} - 14 x + 40$ .
[2 marks]
Answer
(c)
The graph of $y = x^{2} - 14 x + 40$ is reflected in the $x$ -axis. Write down its equation and its turning point.
[2 marks]
Answer $y = \dots\dots\dots\dots$ $(\dots\dots, \dots\dots)$
Total for D2: 6 marks
D3
The point $A (4, 9)$ lies on the graph of $y = f (x)$ . Under the transformation to $y = f (x + p) + q$ , the image of $A$ is $(1, 5)$ . Work out the values of $p$ and $q$ .
[2 marks]
Answer $p = \dots\dots$ $q = \dots\dots$

Extension

Grade 9 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The graph of $y = x^{2}$ is translated by $(p q)$ , then reflected in the $y$ -axis. The result is $y = x^{2} + 10 x + 28$ . Find $p$ and $q$ .
[4 marks]
Hint 1 · What to try
Write the equation after each step.
Hint 2 · The first line
$y = (x - p)^{2} + q$ , then $y = (x + p)^{2} + q$ .
Hint 3 · The full method
$x^{2} + 10 x + 28 = (x + 5)^{2} + 3$ .
Answer $p = \dots\dots$ $q = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the image of a point on a transformed graph.
I can write the equation of a translated or reflected graph.
I can describe fully the transformation between two graphs.

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