Graphing Quadratic Cubic and Exponential Functions

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Graphing Quadratic Cubic and Exponential Functions — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionGraphing Simple Cubic FunctionsConcept Practice

2 marks~3 minCriterion A

An engineer models the vibration of a bridge cable using the cubic function

y = x^3 - 2x

, where

x

represents time (seconds) and

y

represents displacement (metres) from equilibrium. The graph of this function is shown below.

State the number of

x

-intercepts of the graph and the coordinates of the

y

-intercept. [2]

Solutions

2QuestionGraphing and Identifying Exponential Growth and DecayAssessment Practice

4 marks~6 minCriterion D

A wildlife reserve monitors a rare beetle species over four consecutive years. The population is modelled by an exponential function. The graph shows population

P

(in hundreds) against time

t

(in years), with points

(0, 3)

(1, 6)

(2, 12)

, and

(3, 24)

plotted and labelled.

State the initial population of beetles and deduce the growth factor. [2]

Write the equation of the exponential model in the form

P = a \cdot b^t

. [1]

The reserve can sustainably support a maximum of 5000 beetles. Justify whether the reserve manager must intervene before year

t = 5

. [1]

Solutions

3QuestionGraphing Simple Cubic FunctionsConcept Practice

4 marks~6 minCriterion D

A water storage tank is designed so that its volume

V

(in litres) relates to a single length measurement

x

(in metres) by

V = 1000x^3

.

The graph of

y = x^3

is shown below.

State the coordinates of any intercepts of

y = x^3

and describe its end behaviour using the notation

x \to \pm\infty

. [2]

The tank must hold at least 8000 litres. Using the graph of

y = x^3

, justify whether

x = 2

metres meets this design requirement. [2]

Solutions

4QuestionGraphing Simple Cubic FunctionsAssessment Practice

6 marks~9 minCriterion B

A packaging engineer models container volume using

y = x^3

, where

x

is the side length in centimetres and

y

is the volume in cm³. A second design scales volume by a factor of 2, modelled by

y = 2x^3

x

-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3

y = x^3

-27,\ -8,\ -1,\ 0,\ 1,\ 8,\ 27

y = 2x^3

-54,\ -16,\ -2,\ 0,\ 2,\ 16,\ 54

Explain how the

y

-values of

y = 2x^3

relate to those of

y = x^3

, identifying the pattern shown in the table. [2]

Deduce the

y

-values of

y = 3x^3

x = -2

and

x = 4

, using the pattern from part (a) rather than direct calculation. [2]

A third design requires a volume output of at least 150 cm³ when

x = 4

. The engineer proposes

y = ax^3

with

a = 2

. Advise the engineer whether

a = 2

is sufficient and state the minimum integer value of

a

that meets the requirement. [2]

Solutions

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