Dilations and combined transformations

Apply horizontal and vertical dilations to functions; determine the combined effect of dilations and translations on domain, range and asymptotes; identify all transformations from a given equation.

Worked examples

Applying a vertical dilation

Straightforward

Problem

The function

y = \sin x

is dilated vertically by a factor of 2. Write the equation of the new function and state whether this is an enlargement or a reduction.

Combining a dilation and a translation

Moderate

Problem

Starting from

y = \sqrt{x}

, apply (i) a horizontal dilation by factor 4, then (ii) a translation of 1 unit to the left and 2 units up. Write the equation of the resulting function and state its domain and range.

Identifying all transformations from an equation

Challenging

Problem

The equation

y = -2(x + 3)^2 - 1

is obtained from

y = x^2

by a sequence of transformations. Identify all transformations applied, stating the type, direction and magnitude.

Practise

Q1·Straightforward

The function

y = x^2

is dilated vertically by a factor of 3. Which equation represents the transformed function?

$y = 3x^2$$y = (3x)^2$$y = x^2 + 3$$y = x^3$

Q2·Straightforward

The function

y = \sqrt{x}

is transformed to

y = \sqrt{3x}

. What is the scale factor of the horizontal dilation applied?

Q3·Straightforward

A vertical dilation by factor

\dfrac{1}{4}

is applied to

y = x^3

, giving

y = \dfrac{1}{4}x^3

. Evaluate the transformed function at

x = 2

Q4·Straightforward

The function

y = \cos x

is dilated horizontally by a factor of 2, giving

y = \cos\!\left(\dfrac{x}{2}\right)

. Evaluate the transformed function at

x = 2\pi

Q5·Moderate

The function

y = \dfrac{1}{x}

is first dilated horizontally by a factor of 3 and then translated 2 units to the right. What is the

x

-value of the vertical asymptote of the resulting function?

Q6·Moderate

The function

y = 2^x

is dilated vertically by a factor of 4 and then translated 1 unit down. What is the

y

-intercept of the resulting function?

Q7·Moderate

The function

y = \sqrt{x}

(domain

x \geq 0

, range

y \geq 0

) is first dilated vertically by a factor of 5 and then translated 2 units down. What is the lower bound of the range of the resulting function?

Q8·Moderate

The function

y = x^2

is first dilated horizontally by a factor of

\dfrac{1}{2}

, giving

y = (2x)^2 = 4x^2

, and then translated 3 units upward. Evaluate the resulting function at

x = 1

Q9·Challenging

The equation

y = 3f(2x - 6) + 4

is obtained by applying transformations to

y = f(x)

. Rewriting as

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

, how many distinct transformations are applied in total?

Q10·Challenging

The graph of

y = \log_2 x

is transformed to

y = \log_2(4x)

. A point

(a,\, 2)

lies on the original graph. Find the

x

-coordinate of the image of this point on the transformed graph.

Q11·Challenging

The equation

y = (2x - 4)^2 + 1

is a transformation of

y = x^2

. Rewrite the equation by factoring the bracket as

y = (2(x - 2))^2 + 1

, then identify all transformations. What is the

x

-coordinate of the vertex of the transformed parabola?

Q12·Challenging

The function

y = \dfrac{1}{x}

is transformed by a horizontal dilation by factor 2 and then a vertical translation of 3 units up, giving

g(x) = \dfrac{2}{x} + 3

. Find the

x

-value where

g(x) = 4