Transformations of Graphs

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Transformations of Graphs — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionStretching and Compressing GraphsConcept Practice

4 marks~6 minCriterion A

A solar panel manufacturer models the power output (in watts) of a panel using the function

f(x) = x^2

, where

x

is the number of hours after sunrise. A premium panel has greater output, modelled by

g(x) = a\,f(x)

.

The premium panel produces 48 watts exactly 4 hours after sunrise.

Deduce the value of

a

. [2]

Explain what the value of

a

means for the power output of the premium panel compared to the standard panel at every hour of the day. [2]

Solutions

2QuestionCombining Multiple TransformationsAssessment Practice

4 marks~6 minCriterion D

A telecommunications company models the height (in km) of a satellite dish's signal beam using

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

where

x

is the horizontal distance in kilometres from the dish. The parent function is

y = x^2

Describe the sequence of transformations applied to

y = x^2

to obtain

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

. [2]

Explain one mathematical advantage of using this form of the equation to identify the optimal signal coverage position. [1]

The company claims the model is reliable for all distances

x \geq 0

. Advise the company whether this claim should be accepted, supporting your answer with both a mathematical and a physical reason. [1]

Solutions

3QuestionDescribing Graph Changes AlgebraicallyAssessment Practice

7 marks~11 minCriterion C

The diagram shows part of the graph of

y = f(x)

, which passes through the points

A(-2,\, 5)

B(0,\, -3)

, and

C(4,\, 1)

.

The table below shows the coordinates of

A

B

C

and their images under three transformations.

Original on

y = f(x)

(-2,\ 5)

\quad

(0,\ -3)

\quad

(4,\ 1)

Image on

y = f(x + 3)

(-5,\ 5)

\quad

(-3,\ -3)

\quad

(1,\ 1)

Image on

y = f(x - 2)

(0,\ 5)

\quad

(2,\ -3)

\quad

(6,\ 1)

Image on

y = f(x + 5)

(-7,\ 5)

\quad

(-5,\ -3)

\quad

(-1,\ 1)

State the change in the

x

-coordinate of each point when

y = f(x)

is transformed to

y = f(x + 3)

, and state the direction of the resulting shift. [1]

A student claims that the combined transformation

y = f\bigl((x - 2) + 5\bigr)

is equivalent to a single horizontal shift of

y = f(x)

. Find the image of

C(4,\, 1)

under this combined transformation, and hence determine the single equivalent transformation, giving its equation in the form

y = f(x + a)

. [3]

Using the pattern in the table, write a general rule for the horizontal shift produced by

y = f(x + a)

for any real constant

a

. Hence justify why

y = f(x - 5)

shifts the graph of

y = f(x)

exactly

5

units to the right, referring explicitly to at least two rows of the table to support your reasoning. [3]

Solutions

4QuestionStretching and Compressing GraphsAssessment Practice

2 marks~3 minCriterion B

A camera lens models image distortion using

f(x) = x^2

. Two adjusted lens settings produce

g(x) = (2x)^2

and

h(x) = (0.5x)^2

.

Selected coordinate pairs giving equal output values:

f(x)

x

values

1, 2, 3

;

\quad f(x)

values

1, 4, 9

g(x)

x

values

0.5, 1, 1.5

;

\quad g(x)

values

1, 4, 9

h(x)

x

values

2, 4, 6

;

\quad h(x)

values

1, 4, 9

Deduce the general rule for the horizontal transformation of

y = f(x)

when written as

y = f(kx)

, where

k > 0

, by identifying the pattern in the

x

-coordinates across all three functions. [2]

Solutions

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