Worksheets · Higher

Translations and reflections of function graphs

8 exam-style questions, grades 6 to 7. Worked solutions and the marks are on the last page.

Question 1Non-calculator · 4 marks
The curve $y = f(x)$ has a turning point at $(2, -3)$ .
(a) Write down the coordinates of the turning point of $y = f(x) + 4$ . (1)
(b) Write down the coordinates of the turning point of $y = f(x - 3)$ . (1)
(c) Write down the coordinates of the turning point of $y = -f(x)$ . (1)
(d) Write down the coordinates of the turning point of $y = f(-x)$ . (1)
Question 2Calculator · 1 mark
(a) Which transformation maps the graph y = f(x) onto the graph y = f(x $-$ 3)? (1)
1. Translation 3 units left
2. Translation 3 units up
3. Translation 3 units right
4. Translation 3 units down
Question 3Non-calculator · 3 marks
The graph of $y = x^2$ is transformed.
(a) The graph of $y = x^2$ is translated by the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ . Write down the equation of the new graph. (2)
(b) Describe fully the single transformation that maps $y = x^2$ onto $y = (x + 1)^2$ . (1)
1. Translation by $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$
2. Translation by $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$
3. Translation by $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$
Question 4Non-calculator · 3 marks
The point $P(4, 7)$ lies on the curve $y = f(x)$ .
(a) Write down the coordinates of the image of $P$ on the curve $y = f(x) - 5$ . (1)
(b) Write down the coordinates of the image of $P$ on the curve $y = f(x + 1)$ . (1)
(c) Ali says that $(4, 7)$ must also lie on $y = f(-x)$ . Explain why Ali is wrong, and give the point that does. (1)
Question 5Non-calculator · 4 marks
The graph $y=f(x)$ has its unique minimum at $(3,0)$ .
(a) Find the minimum point of $y=f(x-4)+4$ . (2)
(b) Find the maximum point of $y=-f(x)+4$ . (2)
Question 6Non-calculator · 4 marks
The graph $y=f(x)$ has its unique minimum at $(4,1)$ .
(a) Find the minimum point of $y=f(-x)$ . (2)
(b) Find the minimum point of $y=f(-x)-6$ . (2)
Question 7Non-calculator · 4 marks
The graph $y=f(x)$ has its unique minimum at $(5,2)$ .
(a) Find the maximum point of $y=-f(x+6)-8$ . (3)
(b) Explain why the original minimum becomes a maximum. (1)
Question 8Non-calculator · 3 marks
The curve y = f(x) passes through A(−2, 0), B(1, 4) and C(3, 0).
(a) Write down the coordinates of the image of B on the curve y = f(x + 2). (1)
(b) Write down the coordinates of the image of B on the curve y = -f(x). (1)
(c) Write down the solutions of f(x - 1) = 0. (1)

Worked solutions and marks

Question 1

(a) $(2, 1)$

$+4$ outside the function moves the curve up 4: $(2, -3 + 4) = (2, 1)$ .

B1 The correct answer, $(2, 1)$ .

(b) $(5, -3)$

$x - 3$ inside the function moves the curve 3 to the right: $(5, -3)$ .

B1 The correct answer, $(5, -3)$ .

(c) $(2, 3)$

$-f(x)$ reflects the curve in the $x$ -axis: $(2, 3)$ .

B1 The correct answer, $(2, 3)$ .

(d) $(-2, -3)$

$f(-x)$ reflects the curve in the $y$ -axis: $(-2, -3)$ .

B1 The correct answer, $(-2, -3)$ .

Question 2

(a) Translation 3 units right

A point with original input a appears when x $-$ 3 = a.
Its new x-coordinate is a + 3, with the same y-coordinate: a translation 3 units right.

B1 Correct answer: Translation 3 units right

Question 3

(a) $y = (x - 3)^2 - 2$

3 to the right replaces $x$ with $x - 3$ ; 2 down subtracts 2: $y = (x - 3)^2 - 2$ .

B1 $(x - 3)^2$ in the equation.
B1 $y = (x - 3)^2 - 2$ (or $y = x^2 - 6x + 7$ ).

(b) A translation by the vector $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$ .

Adding 1 inside the function moves the graph 1 unit left: a translation by $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$ .

B1 Translation with vector $\begin{pmatrix} -1 \\ 0 \end{pmatrix}$ .

Question 4

(a) $(4, 2)$

Subtracting 5 outside moves every point down 5: $(4, 2)$ .

B1 The correct answer, $(4, 2)$ .

(b) $(3, 7)$

$f(x + 1)$ moves the curve 1 to the left: $(3, 7)$ .

B1 The correct answer, $(3, 7)$ .

(c) $f(-x)$ reflects in the $y$ -axis, so the point is $(-4, 7)$ .

$y = f(-x)$ is the reflection in the $y$ -axis. $P$ maps to $(-4, 7)$ , and $(4, 7)$ is only on it if $f$ is symmetrical about the $y$ -axis.

C1 Explaining that $f(-x)$ reflects in the $y$ -axis, giving $(-4, 7)$ .

Question 5

(a) $(7,4)$

The inside subtraction moves the graph right; the outside addition moves it up.
$x=3+4$
Therefore $(7,4)$ .

M1 The inside subtraction moves the graph right; the outside addition moves it up.
A1 Correct answer: $(7,4)$

(b) $(3,4)$

Reflect the output in the x-axis then translate upwards.
$y=-(0)+4$
Therefore $(3,4)$ .

M1 Reflect the output in the x-axis then translate upwards.
A1 Correct answer: $(3,4)$

Question 6

(a) $(-4,1)$

A negative input reflects the graph in the y-axis.
$x=-4$
Therefore $(-4,1)$ .

M1 A negative input reflects the graph in the y-axis.
A1 Correct answer: $(-4,1)$

(b) $(-4,-5)$

Apply the vertical subtraction after the reflection.
$y=1-6$
Therefore $(-4,-5)$ .

M1 Apply the vertical subtraction after the reflection.
A1 Correct answer: $(-4,-5)$

Question 7

(a) $(-1,-10)$

The inside addition moves the input position left.
$x=5-6$
Reflect vertically and shift down.
$y=-(2)-8$
Therefore $(-1,-10)$ .

M1 The inside addition moves the input position left.
M1 Reflect vertically and shift down.
A1 Correct answer: $(-1,-10)$

(b) Multiplying all output values by −1 reverses their order. The lowest original value becomes the highest reflected value.

Multiplying all output values by −1 reverses their order. The lowest original value becomes the highest reflected value.

C1 Correct conclusion with supporting reasoning: Multiplying all output values by −1 reverses their order. The lowest original value becomes the highest reflected value.

Question 8

(a) $(-1,4)$

Therefore $(-1,4)$ .

B1 Correct answer: $(-1,4)$

(b) $(1,-4)$

Therefore $(1,-4)$ .

B1 Correct answer: $(1,-4)$

(c) $-1, 4$

Therefore $-1, 4$ .

B1 Correct answer: $-1, 4$

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Translations and reflections of function graphs

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