Types of Functions

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Types of Functions — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionQuadratic functionsConcept Practice

4 marks~6 minCriterion A

A landscape architect models the cross-section of a decorative arch using the function

y = x^2

. The design is adjusted for a new location, giving the function

y = (x - 3)^2 + 2

, where

x

and

y

are measured in metres.

Describe the transformations applied to the graph of

y = x^2

to obtain the graph of

y = (x - 3)^2 + 2

. [2]

Deduce the coordinates of the vertex and the equation of the axis of symmetry of

y = (x - 3)^2 + 2

. [1]

The arch must have its lowest point at least 1.5 m above ground level and no more than 4 m horizontally from

x = 0

. Justify whether the adjusted design satisfies both conditions. [1]

Solutions

2QuestionIdentify from graphsAssessment Practice

2 marks~3 minCriterion C

A biologist records two measurements of a growing plant over

x

weeks:

- Plant height (cm), modelled by

y = 2x + 5

- Leaf area (cm²), modelled by

y = x^2 + 3

Identify which equation models plant height and which models leaf area, giving a reason for each. [1]

A classmate claims both measurements could equally well be modelled by a linear function for the first four weeks. Justify whether this claim is valid, and explain what this means for the biologist's conclusions about long-term growth. [1]

Solutions

3QuestionRational functionsAssessment Practice

5 marks~8 minCriterion D

A hospital monitoring system tracks a patient's drug concentration using the model

g(x) = \dfrac{1}{x - 3} + 2

, where

x

is time in hours after administration (

x > 3

) and

g(x)

is concentration in mg/L. The parent function is

f(x) = \dfrac{1}{x}

State the equations of the vertical and horizontal asymptotes of

f(x)

. [1]

Explain how each transformation applied to

f(x)

shifts the asymptotes to produce those of

g(x)

, stating the equation of each asymptote of

g(x)

. [2]

The system triggers a safety alert when concentration remains above 2 mg/L. Justify whether the safety alert ever automatically clears for

x > 3

, using the horizontal asymptote of

g(x)

as the basis for your reasoning. [2]

Solutions

4QuestionQuadratic functionsAssessment Practice

6 marks~9 minCriterion B

A landscape architect models the cross-section of a curved garden bed using a quadratic function

y = ax^2 + bx + c

, where

y

represents the height (cm) of the bed at horizontal position

x

(m).

Measurements taken at equal intervals give:

x

(m):

0, \ 1, \ 2, \ 3, \ 4

y

(cm):

3, \ 6, \ 11, \ 18, \ 27

Calculate the first differences and second differences of the

y

values. [2]

Deduce the values of

a

b

, and

c

, and state the equation of the quadratic function. [2]

The architect requires the garden bed to reach a height of at least 40 cm. Justify whether this condition is met within the measured interval

0 \leq x \leq 4

. [2]

Solutions

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