Problem-solving Pack

Mathematics

Year 11 · 11.5 Transformations of Functions

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

Single transformations. Describe the single transformation that maps the first graph onto the second.

(a)

y = x^2 \to y = (x + 5)^2

(b)

y = x^2 \to y = x^2 + 3

(c)

y = x^2 \to y = -x^2

(d)

y = x^2 \to y = 4x^2

Working space

2Problem 2 of 12

Pinpoint a transformed feature. The graph of

y = f(x)

has a minimum at

(-1, 4)

and

y

-intercept

(0, 7)

.

(a) State the new minimum and

y

-intercept of

y = f(x) - 5

.
(b) State the new minimum and

y

-intercept of

y = f(x + 3)

.
(c) State the new turning point and

y

-intercept of

y = -f(x)

. Is it now a min or max?
(d) State the new minimum and

y

-intercept of

y = f(x/2)

Working space

3Problem 3 of 12

Translation chain. Starting from

y = x^2

:

(a) Translate 2 units right, then 3 units down. Write the equation.
(b) Reflect in the

x

-axis, then translate 1 unit left and 4 units up.
(c) Vertical stretch by 2, then horizontal translation 3 right.

Working space

4Problem 4 of 12

Vertex form to features. Let

f(x) = -2(x + 3)^2 + 5

.

(a) State the vertex and whether it is a max or min.
(b) State the axis of symmetry.
(c) State the

y

-intercept.
(d) Describe how

y = f(x)

is obtained from

y = x^2

as a sequence of transformations.

Working space

5Problem 5 of 12

Inverse chain. Starting from

y = x^2

, the transformations applied are: translate 4 right, vertical stretch by 3, then reflect in the

x

-axis.

(a) Write the final equation.
(b) State a sequence of transformations that returns the final graph to

y = x^2

Working space

6Problem 6 of 12

Find a transformation from data. The graph of

y = f(x)

passes through

(0, 1)

(1, 4)

(2, 13)

. After the transformation

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

, the graph passes through which corresponding points?

(a) State the three points on the graph of

y = g(x)

.
(b) State

g(3)

.
(c) State

g(4)

Working space

7Problem 7 of 12

Stretching. The graph of

y = f(x)

has

x

-intercepts at

x = 1

and

x = 5

, and

y

-intercept

(0, -5)

.

(a) State the corresponding intercepts of

y = 2f(x)

.
(b) State the corresponding intercepts of

y = f(2x)

.
(c) State the corresponding intercepts of

y = -f(x)

Working space

8Problem 8 of 12

Sequence on a sinusoid. Starting from

y = \sin x

:

(a) Write the equation of the graph after a vertical stretch by 3 and a horizontal compression by factor 2.
(b) State its amplitude and period.
(c) Apply the further transformation: translation 1 unit up. Write the new equation.

Working space

9Problem 9 of 12

Match the equation. The graph of

y = f(x)

has key features at

(0, 3)

(2, 0)

and

(4, -3)

. State the equation for the curve that passes through

(0, 6)

(2, 0)

(4, -6)

in terms of

f

Working space

10Problem 10 of 12

Asymptotes and intercepts. The graph of

y = \dfrac{1}{x}

has vertical asymptote

x = 0

and horizontal asymptote

y = 0

.

(a) State the asymptotes of

y = \dfrac{1}{x - 3} + 2

.
(b) Describe the transformations from

y = \dfrac{1}{x}

to that graph.
(c) Find the

y

-intercept of

y = \dfrac{1}{x - 3} + 2

Working space

11Problem 11 of 12

Composing transformations. Let

f(x) = x^2

and

g(x) = -3 f(x - 1) + 4

.

(a) Describe the chain of transformations.
(b) State the vertex of

y = g(x)

and whether it is a max or min.
(c) Find the

y

-intercept of

y = g(x)

Working space

12Problem 12 of 12

Investigation — order of operations. Consider the two chains starting from

y = f(x)

:

- Chain 1: vertical stretch by 3, then translate 2 up.
- Chain 2: translate 2 up, then vertical stretch by 3.

(a) Write the equation of the graph for each chain.
(b) Are they the same? If not, explain why the order matters.
(c) Find a different pair of operations whose order does not matter and justify.

Working space