Linear Functions

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Linear Functions — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionGraphing linesConcept Practice

4 marks~6 minCriterion A

A cyclist begins a journey already some distance from the start of a measured route. The graph of the journey is a straight line passing through

(0, 5)

and

(4, 25)

, where the

x

-axis represents time in hours and the

y

-axis represents distance from the route's start in kilometres.

Calculate the slope of the line and state what it represents in this context. [1]

Deduce the equation of the line in the form

y = mx + c

. [1]

The cyclist's destination is 42 km from the route's start. Using your equation, justify whether the cyclist reaches the destination within 7 hours. [2]

Solutions

2QuestionGraphing linesAssessment Practice

4 marks~6 minCriterion D

A city planner uses the linear model

y = 0.5x + 2

to predict monthly water usage

y

(in megalitres) for a suburb with population

x

(in thousands). Actual usage data for five suburbs is given below.

Suburb A	$x = 10$	actual $= 6.5$ ML
Suburb B	$x = 20$	actual $= 12.0$ ML
Suburb C	$x = 30$	actual $= 17.0$ ML
Suburb D	$x = 40$	actual $= 21.0$ ML
Suburb E	$x = 50$	actual $= 28.0$ ML

Calculate the residual (actual

-

predicted) for each suburb. [2]

Construct a residual plot with population on the horizontal axis. Describe the pattern you observe. [1]

The city planner proposes using this model to set water budgets for all suburbs. Advise the city planner whether this model is sufficient for budget-setting, using your residuals and at least one real-world factor the model ignores. [1]

Solutions

3QuestionSlopeConcept Practice

4 marks~6 minCriterion B

A city's water authority monitors daily water usage at a treatment plant. The graph shows total water processed (in megalitres) against time (in days), modelled as a straight line passing through the labelled points

(1, 2)

and

(4, 8)

Calculate the slope of the line using the two labelled points. [1]

Show that the slope is constant by calculating the rise and run between a different pair of points on the line. [2]

The authority states: "Usage is increasing at a steady rate, so we can reliably forecast demand using this model." Assess whether the mathematical properties of the graph support this statement. [1]

Solutions

4QuestionSlope-intercept formAssessment Practice

4 marks~6 minCriterion C

A mobile data plan charges a fixed monthly fee plus a constant rate per gigabyte used. The total cost, in dollars, is modelled by a linear function. The graph passes through

(0, -4)

and

(3, -2)

Calculate the slope of the line using the two given points. [1]

Deduce the equation of the line in slope-intercept form

y = mx + b

. [1]

A technician claims the monthly fixed fee is a cost saving of 4 dollars. Justify whether this claim accurately describes the overall pricing structure of the plan. [2]

Solutions

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