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Transformations of Functions · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-STranslation, reflection, stretching and compression5 marksPaper 1~8 min

A function

f

is defined by

f(x) = x^2

for

x \in \mathbb{R}

(a)

The graph of

f

is translated by

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

. Find the equation of the resulting function

g

, giving your answer in the form

g(x) = ax^2 + bx + c

. [2 marks]

(b)

The graph of

g

is then reflected in the

x

-axis to give the function

h

. Find the equation of

h

, giving your answer in the form

h(x) = px^2 + qx + r

. [2 marks]

(c)

Write down the range of

h

. [1 mark]

Solutions

2FoundationSAQ-STranslation, reflection, stretching and compression5 marksPaper 1~8 min

The graph of

y = \sqrt{x}

for

x \geq 0

is transformed by a horizontal stretch with scale factor

\dfrac{1}{2}

, followed by a translation by

\begin{pmatrix} 0 \\ 3 \end{pmatrix}

(a)

Find the equation of the resulting function, giving your answer in the form

y = a\sqrt{bx} + c

, where

a

b

, and

c

are integers. [3 marks]

(b)

State the range of the resulting function. [1 mark]

(c)

Find the coordinates of the point on the resulting graph where

x = 4

. [1 mark]

Solutions

3MasterySAQ-STranslation, reflection, stretching and compression6 marksPaper 1~9 min

The function

f

is defined by

f(x) = x^2 - 4x + 7

, for

x \in \mathbb{R}

. The graph of

f

is translated by the vector

\begin{pmatrix} 3 \\ -5 \end{pmatrix}

to obtain the graph of a function

g

(a)

Show that

g(x) = x^2 - 10x + 23

. [3 marks]

(b)

Write down the coordinates of the vertex of the graph of

g

. [1 mark]

(c)

Hence, state the range of

g

and explain why

g

does not have an inverse function for

x \in \mathbb{R}

. [2 marks]

Solutions

4ChallengeSAQ-LTranslation, reflection, stretching and compression7 marksPaper 1~11 min

The function

f

is defined by

f(x) = x^3 - 4x

for

x \in \mathbb{R}

. The function

g

is defined by

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

(a)

State the sequence of three transformations, in the correct order, that maps the graph of

f

onto the graph of

g

. [3 marks]

(b)

The point

P(-1,\, 3)

lies on the graph of

f

. Calculate the coordinates of the image of

P

on the graph of

g

. [2 marks]

(c)

(i) The

x

-intercepts of

f

occur where

f(x) = 0

. Explain why the images of these points under the transformation are not

x

-intercepts of

g

. [1]
(ii) Determine the exact coordinates of the three points on the graph of

g

that correspond to the

x

-intercepts of

f

. [2 marks]

Solutions

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