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Transformations of Functions · Paper 1 — Free Maths AA HL Practice Questions

1FoundationSAQ-STranslation, reflection, stretching, and compressions5 marksPaper 1~8 min

The function

f

is defined by

f(x) = \sqrt{x}

for

x \geq 0

. The graph of

f

is reflected in the

y

-axis and then translated

2

units to the right and

1

unit up to obtain the graph of

g

(a)

Find an expression for

g(x)

, stating its domain. [3 marks]

(b)

The function

h

is defined by

h(x) = g(x)

for

x \leq k

. Determine the value of

k

such that

h^{-1}

exists, and find an expression for

h^{-1}(x)

. [2 marks]

Solutions

2MasterySAQ-STranslation, reflection, stretching, and compressions7 marksPaper 1~11 min

The function

f

is defined by

f(x) = x^2 - 4x + 3

, for

x \in \mathbb{R}

(a)

The graph of

f

is translated by the vector

\begin{pmatrix} 2 \\ -1 \end{pmatrix}

to give the graph of a function

g

. Show that

g(x) = x^2 - 8x + 14

. [2 marks]

(b)

The graph of

g

is reflected in the

y

-axis to give the graph of a function

h

. Find an expression for

h(x)

in the form

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

. [2 marks]

(c)

Find the minimum value of

h

and hence determine whether the graphs of

f

and

h

intersect. [3 marks]

Solutions

3ChallengeSAQ-LTranslation, reflection, stretching, and compressions8 marksPaper 1~12 min

The below shows the graph of

y = f(x)

, where

f

is a quadratic function. The graph has a vertex at

(2, -1)

and passes through the point

(0, 3)

(a)

Determine the equation of

f

in the form

f(x) = a(x - h)^2 + k

. [2 marks]

(b)

The graph of

f

is transformed by: - a reflection in the

x

-axis, - followed by a horizontal stretch with scale factor

2

, - followed by a translation by the vector

\begin{pmatrix} 1 \\ 4 \end{pmatrix}

. The resulting function is

g

. Determine the coordinates of the vertex of the graph of

g

. [3 marks]

(c)

Write

g(x)

in the form

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

, where

p, q, r \in \mathbb{R}

, and hence find the exact

x

-coordinates of the points where the graph of

g

intersects the

x

-axis. [3 marks]

Solutions

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