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

1QuestionInterceptsConcept Practice
4 marks~6 minCriterion A
A city planner models the edge of a new cycle lane using the linear function y=2x−6y = 2x - 6, where xx represents the horizontal distance (metres) from a reference point and yy represents the lane's vertical position. The graph of this function is shown, with both axes clearly labelled and scaled.
a
Explain how to locate the xx-intercept from the graph of y=2x−6y = 2x - 6. [1]
b
Calculate the xx-intercept of y=2x−6y = 2x - 6, showing all algebraic steps. [2]
c
The cycle lane must begin no more than 2 metres from the reference point. Advise the city planner whether this design meets the requirement, justifying your answer using your result from (b). [1]
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2QuestionPlot linear functionsAssessment Practice
6 marks~9 minCriterion C
A student investigates the cooling of a hot liquid by recording its temperature (TT, in °C) every 5 minutes (tt, in minutes):

tt (min): 0, 5, 10, 15, 20, 25

TT (°C): 85, 72, 61, 52, 44, 38

The student proposes the linear model T=−1.8t+85T = -1.8t + 85 to describe the cooling.
a
Calculate the predicted temperature at each time value using the model T=−1.8t+85T = -1.8t + 85. Show all working. [2]
b
Deduce the residual (actual minus predicted) for each data point and describe the pattern you observe. [2]
c
Critique the student's decision to use the linear model T=−1.8t+85T = -1.8t + 85 for this cooling process, with reference to the residuals from part (b) and the real-world behaviour of cooling liquids. [2]
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3QuestionInterpret graphsAssessment Practice
6 marks~9 minCriterion D
A city council uses the quadratic model

h(t)=−4.9t2+20t+2h(t) = -4.9t^2 + 20t + 2

to predict the height (in metres) of a firework shell at time tt seconds. A real test flight shows the shell explodes at t=1.5t = 1.5 s at a height of 1818 m.
a
Calculate h(1.5)h(1.5). [2]
b
The model assumes no air resistance and a perfect launch angle. Evaluate the impact of these assumptions on the model's reliability for safety planning. [2]
c
The council sets a minimum safe clearance height of 2020 m at t=1.5t = 1.5 s. Advise the council whether the model should be used to make this safety decision. [2]
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Solutions

4QuestionPlot quadratic functionsAssessment Practice
6 marks~9 minCriterion B
A structural engineer models the vertical cross-section of a concrete arch using the quadratic function y=ax2+bx+cy = ax^2 + bx + c, where yy is the height (in metres) at horizontal position xx (in metres). Measurements taken at five positions are recorded below.

xx (m): −2, −1, 0, 1, 2-2,\ -1,\ 0,\ 1,\ 2

yy (m): 11, 6, 3, 2, 311,\ 6,\ 3,\ 2,\ 3
a
Calculate the first differences and second differences of the yy-values. [2]
b
Deduce the value of aa, given that the second difference of a quadratic y=ax2+bx+cy = ax^2 + bx + c with unit steps equals 2a2a. Hence state the equation of the arch. [1]
c
The arch must clear a minimum height of 1.51.5 m at x=3x = 3 m to allow vehicles to pass beneath it. Substituting x=3x = 3 into your equation from part (b), advise the engineer whether the arch is safe for vehicles at that position. [3]

Solutions

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