Functions — Free IB MYP5 Mathematics (Extended) Questions

1QuestionDefinition of a functionConcept Practice

2 marks~3 minCriterion C

A school nurse records the ages and heights of students in MYP 5. Two students are both 15 years old; one is 162 cm tall and the other is 178 cm tall.

Let

f: A \to H

represent the mapping from student age

a \in A

(years) to height

h \in H

(cm), where

A = \{14, 15, 16, 17, 18\}

and

H

is the set of plausible student heights in centimetres.

Justify whether

f

qualifies as a function, using the data above as evidence. [2]

Solutions

2QuestionFunction notation f(x)Concept Practice

4 marks~6 minCriterion A

A city's temperature on a winter morning is modelled by the linear function

f(x)

, where

x

is the time in hours after midnight and

f(x)

is the temperature in °C. The graph of

f(x)

passes through

(-1, 4)

and

(3, -2)

.

a

Explain how to read

f(-1)

and

f(3)

directly from the graph, stating both values. [1]

b

Calculate

f(3) - f(-1)

. [1]

c

Interpret the value found in part (b) in the context of the temperature model, and justify whether the temperature is rising or falling over this time interval. [2]

Solutions

3QuestionQuadratic 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.

a

Describe the transformations applied to the graph of

y = x^2

to obtain the graph of

y = (x - 3)^2 + 2

. [2]

b

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

y = (x - 3)^2 + 2

. [1]

c

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

4QuestionRestrictionsConcept Practice

4 marks~6 minCriterion A

A chemical plant monitors the concentration of a coolant in a pipe. The concentration (in grams per litre) at time

x

hours after startup is modelled by

f(x) = \frac{4}{x - 3} + 2, \quad x > 0.

The function has a vertical asymptote at

x = 3

and a horizontal asymptote at

y = 2

.

a

Explain how each asymptote restricts the domain and range of

f

. [2]

b

Calculate

f(5)

. [1]

c

The pipe is safe to operate only when the concentration exceeds 3 g/L. Justify whether the pipe is safe to operate at

x = 5

hours. [1]

Solutions

5QuestionPlot linear functionsConcept Practice

4 marks~6 minCriterion A

A student tests a remote-controlled car on a straight track, recording the following data.

Time (s)	0	1	2	3	4	5
Distance (m)	0	2	4	6	8	10

a

Construct a graph of distance against time, plotting all six points on a fully labelled coordinate grid. Describe the pattern shown by the points. [2]

b

Deduce the equation that models the relationship between distance

d

(m) and time

t

(s), showing your reasoning. [1]

c

The track available for testing is 13 m long. Advise the student whether it is safe to run the car for 7 seconds without reaching the end of the track. Justify your answer with a calculation. [1]

Solutions

6QuestionReflectionsConcept Practice

2 marks~3 minCriterion B

A graphic designer uses coordinate geometry to create a reflected logo. The original logo contains the point

P(3, -5)

. The designer reflects the entire logo across the

x

-axis, as shown by the pattern below.

(2,\ 3) \rightarrow (2,\ -3)

(-1,\ 5) \rightarrow (-1,\ -5)

(4,\ -2) \rightarrow (4,\ 2)

Deduce the general rule for reflecting any point

(x, y)

across the

x

-axis. [1]

State the image of

P(3, -5)

after reflection across the

x

-axis. [1]

Solutions

7QuestionHorizontal shiftsConcept Practice

4 marks~6 minCriterion A

A drone follows a flight path modelled by

f(x) = x^2

, where

x

is horizontal distance (metres) and

f(x)

is height (metres). An adjusted path is modelled by

g(x) = (x-3)^2

.

a

Deduce the coordinates of the vertex of

g

. [1]

b

Calculate

g(7)

and explain what this value represents in the context of the drone's flight path. [2]

c

The drone must reach a height of at least 10 metres by the time it has travelled 6 metres horizontally. Advise the drone operator whether the adjusted path

g

satisfies this requirement, justifying your answer with a calculation. [1]

Solutions

8QuestionSlopeConcept 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)

.

a

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

b

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

c

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

9QuestionPoint-slope formConcept Practice

4 marks~6 minCriterion A

A delivery company charges a fixed connection fee plus a constant rate per kilometre. A driver records that a 1 km trip costs 2 dollars and a 4 km trip costs 8 dollars.

a

Calculate the rate of change (cost per kilometre) using the two data points. [1]

b

Deduce the equation of the cost function in the form

y = mx + b

, where

x

is distance in kilometres and

y

is total cost in dollars. [2]

c

The company advertises "no connection fee." Justify whether this claim is accurate. [1]

Solutions

10QuestionMax/min valuesConcept Practice

3 marks~5 minCriterion A

A sports analyst models the height of a basketball shot using the quadratic function

f(x) = a(x - h)^2 + k

, where

x

is the horizontal distance (metres) from the player and

f(x)

is the height (metres) of the ball.

The graph shows the ball's path. The vertex is at

(0, 6)

and the parabola passes through the point

(2, 2.5)

.

a

State the values of

h

and

k

. [1]

b

Calculate the value of

a

. [1]

c

The analyst claims the ball was released from a height greater than 5 m and that its height decreases on both sides of the launch point. Justify whether this claim is correct, using your values of

a

,

h

, and

k

. [1]

Solutions

11QuestionDecayConcept Practice

4 marks~6 minCriterion A

A radioactive substance decays according to the model

y = 80(0.5)^{t/5}

, where

y

is the mass in grams and

t

is the time in years. The graph passes through the points

(0, 80)

,

(5, 40)

, and

(10, 20)

.

a

Deduce the half-life of the substance from the graph. [1]

b

Calculate the mass remaining after 15 years. [1]

c

A laboratory requires at least 15 g of the substance to conduct a valid experiment. Advise the laboratory whether the experiment can be conducted after 15 years, justifying your answer with reference to your calculated value. [2]

Solutions

12Questionf(g(x)) conceptConcept Practice

2 marks~3 minCriterion A

A construction team is installing a ramp for accessibility compliance. The ramp forms a right triangle: the horizontal base measures

(x + 1)

m and the vertical rise measures

2x

m, where

x = 2

.

Building regulations require that the ramp angle

\theta

(opposite the vertical rise) must not exceed

55°

to be approved.

a

Calculate the value of

\theta

. [1]

b

Advise the construction team whether the ramp should be approved, justifying your answer using your result from part (a). [1]

Solutions

13QuestionEvaluate compositeConcept Practice

2 marks~3 minCriterion C

A delivery company charges a handling fee then applies a distance multiplier. The handling fee function is

f(x) = x + 2

and the distance multiplier is

g(x) = 3x

, where

x

is the base cost in dollars.

a

Calculate

f(g(x))

and

g(f(x))

for

x = 1, 2, 3, 4

. [1]

b

A customer claims the order of operations does not affect the final charge. Justify whether this claim is correct, using your results from part (a). [1]

Solutions

14QuestionPredictionConcept Practice

2 marks~3 minCriterion D

A car travels at a constant speed. A distance–time graph for the journey passes through the origin and the point

(2, 120)

, where distance

d

is measured in kilometres and time

t

is measured in hours.

a

Deduce the equation of the line that models this relationship. [1]

b

The car must complete a 200 km journey within 3 hours. Justify whether the car's speed is sufficient to meet this requirement. [1]

Solutions

15QuestionGraphing constraintsConcept Practice

5 marks~8 minCriterion A

The diagram shows a coordinate grid with the lines

y = 2x + 1

and

x + y = 6

drawn.

a

Write down the coordinates of the point where

y = 2x + 1

crosses the

y

-axis. [1]

b

Find the coordinates of the point of intersection of

y = 2x + 1

and

x + y = 6

. [2]

c

Region

R

satisfies all three inequalities simultaneously:

y \leq 2x + 1, \qquad x + y \leq 6, \qquad x \geq 0.

Justify whether the point

(4, 1)

lies in region

R

. [2]

Solutions

16QuestionDefinition of a functionAssessment Practice

6 marks~9 minCriterion B

A mobile data plan charges a fixed rate per gigabyte used. A network analyst records the following usage data for two customers.

Customer A
Input

x

(GB used):

1,\ 2,\ 3,\ 4

Output

y

(cost in dollars):

3,\ 5,\ 7,\ 9

Customer B
Input

x

(GB used):

1,\ 1,\ 2,\ 3

Output

y

(cost in dollars):

3,\ 4,\ 5,\ 7

a

Analyse the pattern in Customer A's data. Deduce a rule in the form

y = f(x)

and verify it using at least two input-output pairs. [3]

b

Evaluate whether the rule from part (a) applies to Customer B's data, supporting your reasoning with calculations. [1]

c

The analyst must decide whether to use a single function to model Customer B's pricing. Justify whether or not this is appropriate, and explain what this means for the reliability of the pricing model for Customer B. [2]

Solutions

17QuestionEvaluating functionsAssessment Practice

2 marks~3 minCriterion D

A small business models its weekly production cost using

C(x) = 5x + 20

, where

x

is the number of items produced and

C(x)

is the cost in dollars.

a

Calculate

C(10)

and

C(25)

. [1]

b

The business considers scaling production to 1000 items per week. Justify whether the linear model

C(x) = 5x + 20

is a reliable predictor of costs at this scale. [1]

Solutions

18QuestionIdentify 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

a

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

b

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

19QuestionIdentify from graphsAssessment Practice

4 marks~6 minCriterion B

A scientist records the spread of a ground-cover plant (in cm²) over four consecutive weeks. Four candidate models are proposed; their graphs are labelled A–D.

Graph A:

(-2,\ -4),\ (-1,\ -2),\ (0,\ 0),\ (1,\ 2),\ (2,\ 4)

Graph B:

(-2,\ 4),\ (-1,\ 1),\ (0,\ 0),\ (1,\ 1),\ (2,\ 4)

Graph C:

(-2,\ 0.25),\ (-1,\ 0.5),\ (0,\ 1),\ (1,\ 2),\ (2,\ 4)

Graph D:

(-2,\ 2),\ (-1,\ 1),\ (0,\ 0),\ (1,\ 1),\ (2,\ 2)

a

Calculate the first differences for each graph. [1]

b

Deduce which graph models linear growth and which models quadratic growth. Justify your answer using first and second differences. [2]

c

The scientist needs a model predicting accelerating, unbounded growth with no reflective symmetry about the vertical axis. Recommend which of the four graphs best fits this requirement, and justify your choice using the pattern of its differences. [1]

Solutions

20QuestionRational 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}

.

a

State the equations of the vertical and horizontal asymptotes of

f(x)

. [1]

b

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]

c

The system triggers a safety alert when concentration remains above 2 mg/L. Using the horizontal asymptote of

g(x)

, advise whether the safety alert will ever automatically clear for

x > 3

. Justify your answer. [2]

Solutions

21QuestionRange from graphsAssessment Practice

4 marks~6 minCriterion B

A structural engineer models the vertical clearance beneath four cable-stay bridge sections using quadratic functions. Each function gives height

y

(metres) at horizontal position

x

(metres), with the vertex representing the lowest or highest point of the cable profile.

The four cable profiles are:

f(x) = x^2

, vertex

(0,\ 0)

, opens upward

g(x) = x^2 + 3

, vertex

(0,\ 3)

, opens upward

h(x) = (x-2)^2

, vertex

(2,\ 0)

, opens upward

k(x) = -(x+1)^2 + 4

, vertex

(-1,\ 4)

, opens downward

a

State the range of each function. [1]

b

Analyse the relationship between the vertex

y

-coordinate, the direction of opening, and the range of each profile. Deduce a general rule connecting the sign of

a

and the vertex

y

-coordinate

k

to the range of

a(x-h)^2 + k

. [2]

c

A new bridge section requires a cable profile

m(x) = a(x-h)^2 + k

whose height never exceeds

-3

m relative to the reference datum, so the range must be

(-\infty,\ -3]

. Construct a valid function

m(x)

satisfying this condition, and justify whether a cable profile with range

(-\infty,\ -3]

is physically safe for vehicles passing beneath the bridge. [1]

Solutions

22QuestionRange from graphsAssessment Practice

3 marks~5 minCriterion D

A drone delivery company models the height of a package above ground during a short flight using the function

h(t) = t^2 - 4t + 3

, where

h

is the height in metres and

t

is the time in seconds. The graph of

h(t)

shows a parabola with vertex

(2, -1)

that opens upwards.

a

Deduce the range of

h(t)

. [1]

b

The package must remain at or above ground level (

h \geq 0

) throughout the flight. Justify whether this model is physically valid, using your answer to part (a). [2]

Solutions

23QuestionRange from graphsAssessment Practice

5 marks~8 minCriterion C

A hospital monitoring system records a patient's oxygen saturation level,

f(x)

(%), over a 24-hour period. The piecewise function

f(x)

is represented by the graph below.

The graph consists of three parts:
- a horizontal segment at

y = 4

for

-5 \leq x < -2

, with a closed circle at

(-5, 4)

and an open circle at

(-2, 4)

- a downward-opening parabola with vertex

(0, 3)

, passing through

(-2, 1)

and

(2, 1)

, with closed circles at both endpoints
- a curve for

x > 2

starting at

(2, 1)

with a closed circle, decreasing toward the asymptote

y = -1

as

x \to \infty

, never reaching

y = -1

a

State the

y

-value produced by the horizontal segment and identify whether it is included in the range. [1]

b

Deduce the set of

y

-values contributed by the parabola and by the asymptotic curve, showing your reasoning for each endpoint. [2]

c

Interpret the complete range of

f(x)

in interval notation, and justify whether the gap in the range gives cause for clinical concern about the reliability of the monitoring data. [2]

Solutions

24QuestionPlot 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 = ax^2 + bx + c

, where

y

is the height (in metres) at horizontal position

x

(in metres). Measurements taken at five positions are recorded below.

x

(m):

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

y

(m):

11,\ 6,\ 3,\ 2,\ 3

a

Calculate the first differences and second differences of the

y

-values. [2]

b

Deduce the value of

a

, given that the second difference of a quadratic

y = ax^2 + bx + c

with unit steps equals

2a

. Hence state the equation of the arch. [1]

c

The arch must clear a minimum height of

1.5

m at

x = 3

m to allow vehicles to pass beneath it. Substituting

x = 3

into your equation from part (b), advise the engineer whether the arch is safe for vehicles at that position. [3]

Solutions

25QuestionPlot linear functionsAssessment Practice

6 marks~9 minCriterion C

A student investigates the cooling of a hot liquid by recording its temperature (

T

, in °C) every 5 minutes (

t

, in minutes):

t

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

T

(°C): 85, 72, 61, 52, 44, 38

The student proposes the linear model

T = -1.8t + 85

to describe the cooling.

a

Calculate the predicted temperature at each time value using the model

T = -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 + 85

for this cooling process, with reference to the residuals from part (b) and the real-world behaviour of cooling liquids. [2]

Solutions

26QuestionPlot quadratic functionsAssessment Practice

4 marks~6 minCriterion D

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

y = (x - 3)^2 + 2

, where

x

and

y

are measured in metres. The parent function is

y = x^2

.

a

State the coordinates of the vertex of

y = (x - 3)^2 + 2

and write down the equation of its axis of symmetry. [2]

b

Explain the two transformations that map the graph of

y = x^2

onto the graph of

y = (x - 3)^2 + 2

. [1]

c

The arch must clear a pathway centred at

x = 3

that is 4 metres wide and 2.5 metres tall. Advise the architect whether the arch satisfies both clearance requirements, supporting your advice with calculations. [1]

Solutions

27QuestionStretch/compressionAssessment Practice

6 marks~9 minCriterion C

A graphic designer is resizing a logo for a billboard and a business card. The logo's outline is defined by vertices

A(2,\ 1)

,

B(6,\ 1)

,

C(7,\ 4)

, and

D(1,\ 4)

. The designer applies a vertical stretch by factor

2.5

, followed by a horizontal compression by factor

0.4

.

a

Determine the coordinates of the transformed vertices

A'

,

B'

,

C'

, and

D'

after both transformations are applied in the given order. [2]

b

Calculate the area of the original logo and the area of the transformed logo. Show that the two areas are equal, and explain why this result occurs. [2]

c

Advise the designer whether this transformation is suitable for both media, justifying your answer with reference to the effect of non-uniform scaling on the logo's proportions and the significance of the equal-area result. [2]

Solutions

28QuestionCombined transformationsAssessment Practice

2 marks~3 minCriterion D

A graphic designer uses a triangle with vertices at

A(1, 2)

,

B(3, 2)

, and

C(2, 5)

. She first translates the triangle by the vector

\begin{pmatrix} 4 \\ -1 \end{pmatrix}

, then reflects the resulting triangle across the vertical line

x = 6

.

Justify whether this mathematical model is reliable for a designer working with a physical stencil on a printed page. [2]

Solutions

29QuestionSlopeAssessment Practice

4 marks~6 minCriterion D

A distance–time graph shows a cyclist's journey from A to D, with points A(0, 0), B(20, 5), C(60, 5), and D(80, 20), where time is in minutes and distance is in kilometres.

a

Show that the slope of segment AB is 0.25 km/min. [1]

b

Explain what the slope of segment AB represents in the context of the cyclist's journey. [1]

c

The cyclist must maintain an average speed of at least 0.20 km/min over the entire journey from A to D. Justify whether this condition is met. [2]

Solutions

30QuestionParallel & perpendicularAssessment Practice

2 marks~3 minCriterion C

A city planner is designing two straight pedestrian pathways that must cross at a right angle to ensure safe sightlines at the intersection.

The first pathway has a slope of

m_1 = \dfrac{2}{3}

.

a

Deduce the slope of the second pathway. [1]

b

The planner states: "A steeper crossing angle improves pedestrian safety." Justify whether the slopes of the two pathways satisfy the planner's safety condition. [1]

Solutions

31QuestionAxis of symmetryAssessment Practice

5 marks~8 minCriterion B

A skateboard ramp is modelled by

y = (x - p)^2 + q

, where

x

is the horizontal distance (metres) from a reference point and

y

is the height (metres). The axis of symmetry gives the horizontal position of the ramp's lowest point.

a

Calculate the axis of symmetry for each function. [1]

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

,

\quad y = (x-(-1))^2 - 4

,

\quad y = (x-0)^2 + 5

,

\quad y = (x-2)^2 + 1

,

\quad y = (x-5)^2 - 3

Axis of symmetry: ___ , ___ , ___ , ___ , ___

b

Deduce the general rule for the axis of symmetry of

y = (x - p)^2 + q

. [1]

c

A ramp is modelled by

y = (x + 4)^2 - 7

. Use your rule to determine the horizontal position of the ramp's lowest point, showing full working and verifying your answer algebraically. [2]

d

The ramp's lowest point must lie between

x = -5

and

x = -3

to fit safely within the skate park. Justify whether this ramp meets the safety requirement. [1]

Solutions

32QuestionStandard formAssessment Practice

4 marks~6 minCriterion C

A road engineer models the cross-section of a speed bump using the function

f(x) = 2x^2 - 8x + 6

, where

x

is the horizontal distance in metres from a reference point and

f(x)

is the height in metres. The graph of

f(x)

is provided.

a

State the coordinates of the

y

-intercept of

f(x)

. [1]

b

Find the

x

-intercepts of

f(x)

by solving

f(x) = 0

. [1]

c

A colleague proposes the alternative model

g(x) = x^2 - 4x + 3

. Justify whether

f(x)

or

g(x)

produces the steeper speed-bump profile, and advise the engineer which model to select if road safety regulations require the steepest possible cross-section. [2]

Solutions

33QuestionVertex formAssessment Practice

8 marks~12 minCriterion D

A drone's GPS logger records the following flight data:

Time (

t

in seconds): 0, 1, 2, 3, 4

Height (

h(t)

in metres): 5, 25, 35, 35, 25

The flight path is modelled as a parabola.

a

Construct the quadratic function in vertex form

h(t) = a(t - p)^2 + q

that models this data. [2]

b

A height restriction of 40 metres applies in this flight zone. Deduce, using your model, whether the drone complies with this restriction. [2]

c

The model assumes a perfect parabolic path with no wind or air resistance. Advise the drone operator whether the model's compliance conclusion from part (b) alone is sufficient to authorise the flight, justifying your answer with reference to the model's limitations. [4]

Solutions

34QuestionBase interpretationAssessment Practice

4 marks~6 minCriterion C

Two bacteria strains are monitored in a laboratory. Strain A doubles every hour, modelled by

y = 2^x

, and Strain B triples every hour, modelled by

y = 3^x

, where

x

is time in hours.

a

Calculate the population of each strain at

x = 2

hours. [1]

b

Deduce the ratio of Strain B's population to Strain A's population at

x = 2

, and show how this ratio relates to the bases of the two functions. [2]

c

A lab protocol requires Strain B's population to be more than twice that of Strain A at all monitoring points. Justify whether this protocol should be considered reliable for all

x \geq 1

. [1]

Solutions

35QuestionGraphing exponentialAssessment Practice

2 marks~3 minCriterion B

The graph shows two exponential functions: curve A,

y = 2^x

, and curve B,

y = \left(\dfrac{1}{2}\right)^x

.

Explain why the two curves are reflections of each other across the

y

-axis, using the relationship between the two functions. In your answer, state what this means for a general point

(a,\, b)

on curve A. [2]

Solutions

36QuestionDecayAssessment Practice

6 marks~9 minCriterion D

A patient receives a single 200 mg dose of a drug. The drug decays exponentially, with 88% remaining after each hour. The minimum effective concentration is 25 mg.

a

Construct an exponential decay model for the amount of drug

A(t)

remaining after

t

hours, in the form

A(t) = A_0\,e^{-kt}

, showing how you determine

k

. [2]

b

Deduce the hour at which the drug level first falls below 25 mg. [2]

c

Advise a clinician whether this model alone is sufficient to guide a repeat-dosing schedule, discussing at least two limitations and their implications for patient safety. [2]

Solutions

37QuestionEvaluate compositeAssessment Practice

6 marks~9 minCriterion B

A computer simulation models the spread of a notification through a network. At each step, every active node activates exactly one new node and remains active. The number of active nodes at step

n

is modelled by

f(x) = 2x + 1

, applied repeatedly from

x_1 = 1

. The sequence is defined by

x_{k+1} = f(x_k)

for all positive integers

k

.

a

Calculate the first five terms of the sequence. [2]

b

Deduce a formula for

x_n

in terms of

n

, showing clearly how the pattern in part (a) leads to your general expression. [2]

c

Prove by mathematical induction that your formula holds for all positive integers

n

. [2]

d

The network has a capacity of 1000 active nodes. Using your formula, advise the network administrator at which step they should intervene to prevent the capacity being exceeded, justifying your answer with reference to the simulation. [2]

Solutions

38QuestionInterpret compositeAssessment Practice

2 marks~3 minCriterion D

A toy factory models the painting cost

C

(in dollars) of a car as a function of its surface area

A

(in cm²), and the surface area as a function of a single dimension

d

(in cm), such that

A(d) = 6d^2

and

C(A) = 0.05A

.

a

Explain how the composite function

C(A(d))

models the total painting cost in terms of dimension

d

, showing the chain of dependence

d \rightarrow A \rightarrow C

. [1]

b

Interpret what a doubling of dimension

d

means for the painting cost, and justify whether this composite model is sufficient for a factory manager to use when pricing larger car models. [1]

Solutions

39QuestionReal-life modelsAssessment Practice

4 marks~6 minCriterion B

A student compares two savings plans. The total saved (in dollars) at the end of each week is recorded below.

Week (

n

): 1, 2, 3, 4, 5

Plan A (

S

): 50, 100, 150, 200, 250

Plan B (

S

): 50, 105, 170, 245, 330

a

Deduce the type of pattern (linear or quadratic) for each plan and derive a formula for total savings

S

after

n

weeks. [2]

b

Construct a prediction for the total savings under each plan after 10 weeks. [1]

c

Advise the student which savings plan to follow over a 52-week year, supporting your recommendation with calculated evidence. [1]

Solutions

40QuestionInterpret contextAssessment Practice

6 marks~9 minCriterion A

A taxi company charges a fixed flag-fall fee plus a constant rate per kilometre travelled. The table below shows sample fares.

Distance (km)	1	3	5
Total cost (dollars)	7.50	14.50	21.50

a

Deduce the rate of change and the flag-fall fee, and hence write the general formula for the total cost

C

(in dollars) in terms of distance

d

(in km). [3]

b

A rival company charges according to

C = 4.20d + 2.00

. Construct a calculation to determine the distance at which both companies charge the same fare. [1]

c

A passenger must travel 8 km and has exactly 32.00 dollars. Advise the passenger which company to use, justifying your recommendation with numerical evidence. [2]

Solutions

41QuestionPredictionAssessment Practice

6 marks~9 minCriterion D

A projectile is launched at 10 m/s. The maximum height

H

(in metres) reached at launch angle

\theta

is given by

H = \frac{(10\sin\theta)^2}{2g}, \quad g = 9.8 \text{ m/s}^2

Angle

\theta

(degrees): 15, 30, 45, 60, 75

Maximum height

H

(m): ?, ?, ?, ?, ?

a

Calculate

H

for each angle and complete the table. Round to two decimal places. [2]

b

Explain how

H

changes as

\theta

increases from 15° to 75°, and identify the mathematical relationship between

H

and

\theta

. [2]

c

A sports engineer claims that launching at 90° gives the greatest possible height. Advise the engineer whether a 90° launch angle is appropriate for a projectile intended to cover horizontal distance. [2]

Solutions

42QuestionCorner pointsAssessment Practice

6 marks~9 minCriterion B

The diagram shows the feasible regions for three systems of inequalities.

System A:

x \ge 0

,

y \ge 0

,

x + y \le 6

System B:

x \ge 0

,

y \ge 0

,

x + y \le 6

,

x \le 4

System C:

x \ge 0

,

y \ge 0

,

x + y \le 6

,

x \le 4

,

y \le 3

a

Write down the coordinates of every corner point of the feasible region for each system. [2]

b

Describe the pattern connecting the number of constraints to the number of corner points across Systems A, B, and C. [1]

c

A student adds the constraint

y \le 1

to System C to create System D. Justify whether the pattern identified in part (b) still holds for System D, identifying any new corner point(s) and explaining why the corner-point count changes by the amount it does. [3]

Solutions

43QuestionCorner pointsAssessment Practice

8 marks~12 minCriterion C

The feasible region of a linear programming problem is bounded by the constraints:

y \geq 0, \quad x \geq 0, \quad y \leq -x + 8, \quad y \leq -\tfrac{1}{2}x + 5

The diagram shows the feasible region with three of its corner points labelled:

A(0, 0)

,

B(0, 5)

, and

D(8, 0)

.

a

Find the coordinates of the fourth corner point

C

, where the lines

y = -x + 8

and

y = -\tfrac{1}{2}x + 5

intersect. [2]

b

An objective function is

P = 3x + 4y

. Evaluate

P

at each of the four corner points and hence state the coordinates of the corner point that gives the maximum value of

P

, giving the maximum value. [3]

c

A student claims: "The maximum of a linear objective function over a polygonal feasible region always occurs at a corner point, because the level lines

P = k

are parallel to each other and the last one to touch the region must do so at a vertex." Justify whether this reasoning is mathematically correct, referring to the gradient of the objective function's level lines and the geometry of the feasible region. [3]

Solutions

44QuestionInequalitiesAssessment Practice

10 marks~15 minCriterion D

A small business produces wooden chairs and tables. Each chair requires 2 hours of labour and 1 unit of wood. Each table requires 3 hours of labour and 2 units of wood. The business has at most 240 hours of labour and 160 units of wood available per week. The profit per chair is 40 dollars and per table is 60 dollars.

a

Define your variables and write down all constraints as inequalities. [2]

b

Construct the feasible region on the axes provided, labelling all boundary lines and vertices. [3]

c

Determine the production plan that maximises weekly profit, stating the maximum profit. [2]

d

Advise the business owner whether this linear programming model should be used to make weekly production decisions, identifying one limitation that affects its reliability. [3]

Solutions

Functions — Free MYP5 Mathematics (Extended) Practice Questions