Functions — Free IB MYP4 Mathematics (Extended) Questions

1QuestionFunction notation f(x)Concept Practice

2 marks~3 minCriterion A

A structural engineer models the hypotenuse of a right-triangular roof truss using the function

f(x) = 3x + 5

, where

x

is a design parameter in metres and

f(x)

gives the hypotenuse length in centimetres. Building regulations require the hypotenuse to be no longer than 20 cm for this truss type.

a

Calculate

f(4)

. [1]

b

Justify whether the truss meets the building regulation when

x = 4

. [1]

Solutions

2QuestionLinear functionsConcept Practice

4 marks~6 minCriterion A

A cyclist's journey is shown on the distance–time graph below. The two labelled points are

(1, 15)

and

(3, 45)

, where the horizontal axis shows time in hours and the vertical axis shows distance in kilometres.

a

Interpret what the gradient of the line represents in the context of this journey. [1]

b

Calculate the cyclist's speed in km/h. [1]

c

A cycling event requires competitors to cover at least 40 km in the first 2.5 hours. Justify whether this cyclist meets that requirement. [2]

Solutions

3QuestionDomain from graphsConcept Practice

3 marks~5 minCriterion A

A solar panel technician monitors the power output of a panel. The output is modelled by the function

P = 3t - 1

, where

P

is the power output in watts and

t

is the time in hours after sunrise. The function is valid for

2 \leq t \leq 8

.

a

Calculate the minimum and maximum values of

P

for this domain. [2]

b

The panel must deliver at least 20 watts to charge a battery. Justify whether this condition is ever met within the given domain. [1]

Solutions

4QuestionInterceptsConcept Practice

4 marks~6 minCriterion A

A city planner models the edge of a new cycle lane using the linear function

y = 2x - 6

, where

x

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

y

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

x

-intercept from the graph of

y = 2x - 6

. [1]

b

Calculate the

x

-intercept of

y = 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]

Solutions

5QuestionReflectionsConcept 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

6QuestionStretch/compressionConcept Practice

4 marks~6 minCriterion A

A photographer uses editing software to adjust image brightness. The software models the original brightness of a pixel as

f(x) = x^2

, where

x

is the input light value. The adjusted brightness is

g(x) = a \cdot f(x)

.

The point

(2, 4)

lies on

f

and the point

(2, 8)

lies on

g

.

a

Deduce the value of

a

. [1]

b

Explain what the vertical stretch by factor

a

means for how the software changes pixel brightness. [1]

c

A pixel has input value

x = 5

. The software requires adjusted brightness to exceed 60 to display correctly. Justify whether this pixel displays correctly after the transformation. [2]

Solutions

7QuestionGraphing 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.

a

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

b

Deduce the equation of the line in the form

y = mx + c

. [1]

c

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

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

9QuestionBase interpretationConcept Practice

2 marks~3 minCriterion A

A population of bacteria in a laboratory experiment is modelled by

y = a \cdot b^x

, where

y

is the number of bacteria (in thousands) and

x

is the time in hours. The graph of this model passes through the points

(0, 3)

and

(1, 6)

.

a

Deduce the value of

a

. [1]

b

The laboratory considers the experiment viable only if the initial bacterial population exceeds 2500. Justify whether this condition is met. [1]

Solutions

10QuestionEvaluate 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

11Questionf(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

12QuestionInterpret contextConcept Practice

2 marks~3 minCriterion A

A bicycle rental company charges a fixed registration fee plus an hourly rate. The total cost

C

(in dollars) for renting a bicycle for

h

hours is recorded below.

Hours ( $h$ )	1	2	3	4
Total cost ( $C$ in dollars)	8	11	14	17

a

Deduce the equation for

C

in terms of

h

. [1]

b

Interpret what the fixed registration fee and the hourly rate represent in this context, and identify which variable is dependent and which is independent. [1]

Solutions

13QuestionCorner pointsConcept Practice

5 marks~8 minCriterion A

The diagram shows the feasible region defined by the constraints

x \ge 0

,

y \ge 0

,

x + y \le 6

, and

2x + y \le 10

, with corner points labelled

A

,

B

,

C

, and

D

.

a

Find the coordinates of corner point

C

, the intersection of

x + y = 6

and

2x + y = 10

, showing a complete algebraic method. [2]

b

An objective function is defined as

P = 3x + 5y

. Calculate the value of

P

at each of the four corner points

A(0,0)

,

B(5,0)

,

C

, and

D(0,6)

, and hence justify which corner point maximises

P

, stating the maximum value. [3]

Solutions

14QuestionDefinition 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

15QuestionRelation vs functionAssessment Practice

4 marks~6 minCriterion C

A parabolic solar reflector is designed so that its cross-section satisfies the relation

x = y^2 - 2

, where

x

and

y

are measured in metres.

a

Determine the domain and range of this relation. [2]

b

Explain whether this relation represents a function, using the vertical line test. [1]

c

A sensor must be placed at a unique horizontal position for each height

y

. Justify whether the relation

x = y^2 - 2

, as stated, is suitable for this purpose, and state any restriction needed. [1]

Solutions

16QuestionEvaluating functionsAssessment Practice

4 marks~6 minCriterion D

A mobile data plan charges a monthly fee. The total cost, in dollars, is modelled by

f(x) = 2x + 1

, where

x

is the number of gigabytes used.

The graph of

f

is shown, with points marked at

x = -1

,

x = 0

, and

x = 2

.

a

State the coordinates of the three marked points. [1]

b

Show that

f(4) = 9

. [1]

c

Explain what

f(4) = 9

means in this context, and justify whether the linear model is appropriate for this situation. [2]

Solutions

17QuestionIdentify from equationsAssessment Practice

4 marks~6 minCriterion B

A landscape architect is designing square garden plots. Each plot has a side length of

x

cm, and the usable planting area

y

cm² follows the pattern below.

x

(cm):

1, \ 2, \ 3, \ 4

y

(cm²):

3, \ 7, \ 13, \ 21

a

Calculate the first differences and second differences for the

y

-values. [2]

b

Deduce the type of function (linear or quadratic) that models this data, justifying your answer using the differences found in part (a). [1]

c

The architect needs the usable planting area to exceed 90 cm². Advise the architect whether a plot with side length

x = 6

cm is sufficient to meet this requirement. [1]

Solutions

18QuestionLinear functionsAssessment Practice

8 marks~12 minCriterion C

A logistics company purchases a delivery van for 30 000 dollars. The van depreciates at a constant rate of 4 000 dollars per year.

a

Construct a linear function

V(t)

that represents the value of the van, in dollars, after

t

years. [2]

b

Deduce the value of the van after 3 years. Show all working. [2]

c

The company's accountant states: "We should sell the van before its value falls below 14 000 dollars." Determine after how many years this threshold is reached, then advise the accountant whether the van should be sold before or after the 4-year mark. Show all working. [4]

Solutions

19QuestionRational functionsAssessment Practice

3 marks~5 minCriterion D

A community centre charges a fixed booking fee of 250 dollars and an additional 15 dollars per person for catering. The average cost per person for

x

guests is modelled by

A(x) = \frac{250}{x} + 15.

a

Calculate the average cost per person when there are 50 guests. [1]

b

Explain what happens to the average cost per person as the number of guests increases, linking your answer to the structure of

A(x)

. [1]

c

The community centre manager claims the model is reliable for planning any event. Advise the manager on whether this claim is justified, identifying one real-world constraint that limits the model. [1]

Solutions

20QuestionDomain from graphsAssessment Practice

6 marks~9 minCriterion D

The height

h

(in metres) of a projectile fired from the ground is modelled by

h(t) = -5t^2 + 40t

, where

t

is time in seconds. The graph shows a downward-opening parabola with vertex

(4,\ 80)

, crossing the

t

-axis at

t = 0

and

t = 8

. The horizontal axis shows

t

from 0 to 9 seconds; the vertical axis shows

h

from 0 to 90 metres.

a

State the domain of

h(t)

in this context. [1]

b

Using the graph, estimate the two values of

t

at which the projectile reaches a height of 75 m. [2]

c

A sensor records the projectile's height as 46.25 m at

t = 1.5

s. Calculate the percentage error between the model's prediction and the sensor reading, using

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{|\text{actual}|} \times 100\%.

[2]

d

The safety system requires the model to be accurate to within 5\% at all times. Justify whether the model satisfies this requirement at

t = 1.5

s. [1]

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

22QuestionDomain from graphsAssessment Practice

4 marks~6 minCriterion C

A city's water pressure monitoring system records pressure (in bar) as a piecewise linear function of time (in hours) over a 24-hour period. Due to scheduled maintenance, no data is recorded between

t = -1

and

t = 1

.

The graph shows two linear segments:
- Segment 1: from

(-4,\ 2)

(closed circle) to

(-1,\ 5)

(open circle)
- Segment 2: from

(1,\ 3)

(closed circle) to

(4,\ 0)

(closed circle)

a

Justify whether this graph represents a function. [1]

b

Deduce the domain of the function, expressing your answer in interval notation. [1]

c

The monitoring system requires at least 7 hours of combined recorded data to be considered reliable. Advise the network operator whether this session meets the reliability requirement, supporting your answer with calculations. [2]

Solutions

23QuestionPlot 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

24QuestionInterpret graphsAssessment Practice

6 marks~9 minCriterion D

A city council uses the quadratic model

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

to predict the height (in metres) of a firework shell at time

t

seconds. A real test flight shows the shell explodes at

t = 1.5

s at a height of

18

m.

a

Calculate

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

20

m at

t = 1.5

s. Advise the council whether the model should be used to make this safety decision. [2]

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

26QuestionHorizontal shiftsAssessment Practice

6 marks~9 minCriterion D

Seismologists model the P-wave arrival time

t

(seconds) at a recording station using

t = \frac{d}{v} + h

where

d

is the distance (km) from the epicentre,

v

is the wave speed (km/s), and

h

is the station's clock offset (seconds).

Station A is 120 km from an epicentre. The wave speed is

v = 6

km/s and the clock offset is

h = 0.5

s.

Station B is 200 km from the same epicentre, with

v = 6

km/s. Its recorded arrival time is

34.2

s and its clock offset

h_B

is unknown.

a

Calculate the predicted arrival time

t_A

at Station A. [2]

b

Determine the clock offset

h_B

at Station B. [2]

c

Clock offsets carry a

\pm 0.5

s uncertainty and wave speed is assumed constant across all rock layers. Advise a seismologist whether this model is sufficiently reliable to locate the epicentre for emergency response purposes. [2]

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

28QuestionGraphing 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

a

Calculate the residual (actual

-

predicted) for each suburb. [2]

b

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

c

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

29QuestionSlope-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)

.

a

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

b

Deduce the equation of the line in slope-intercept form

y = mx + b

. [1]

c

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

30QuestionAxis 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

31QuestionGraphing parabolasAssessment Practice

4 marks~6 minCriterion C

A sports analyst models the height of a basketball shot using the equation

y = 2(x - 3)^2 - 8

, where

x

is the horizontal distance (metres) from the player and

y

is the height (metres) relative to the release point. The graph of this parabola is shown.

a

State the coordinates of the vertex of the parabola. [1]

b

Calculate the

x

-intercepts of the parabola. Show your working. [2]

c

The analyst claims the ball lands on the ground at two points either side of the player. Justify whether this claim is valid, using your results from part (b) and the context. [1]

Solutions

32QuestionAxis of symmetryAssessment Practice

6 marks~9 minCriterion A

A water fountain projects a jet of water from a nozzle at ground level. The path is modelled by

h(x) = ax^2 + bx + c

, where

h

is the height (m) and

x

is the horizontal distance (m). The nozzle is at

x = 0

and the water lands at

x = d

. The designer wants to maximise the horizontal distance

d

.

a

State the axis of symmetry of

h(x) = ax^2 + bx + c

and explain what it represents in this context. [2]

b

The nozzle and landing point are both at ground level. Using the axis of symmetry, deduce that the maximum horizontal distance requires

-\dfrac{b}{2a} = \dfrac{d}{2}

. [2]

c

A wall stands at

x = 3

m with height 4 m. The designer proposes

h(x) = -x^2 + 6x

. Evaluate whether this proposal should be accepted. [2]

Solutions

33QuestionVertex formAssessment Practice

2 marks~3 minCriterion D

A basketball free throw is modelled by

h(t) = -5(t - 1)^2 + 4

where

h

is the height of the ball in metres and

t

is the time in seconds after release.

a

State the vertex of the parabola and interpret its meaning in this context. [1]

b

The rim of the basket is at a height of 3.05 m. Determine the two values of

t

at which the ball is at rim height, then justify which value corresponds to a successful free throw. [1]

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

8 marks~12 minCriterion D

A pharmaceutical company models the concentration of a drug in a patient's bloodstream as

C(t) = 200 \cdot (0.85)^t

mg/L, where

t

is the time in hours after injection. A nurse records the following measured concentrations:

Time (hours)	1	2	4	6
Measured concentration (mg/L)	170	144	104	75

a

Calculate the predicted concentration at each time point using the model. Present your results alongside the measured values. [2]

b

Calculate the percentage error at each time point using

\% \text{ error} = \frac{|\text{predicted} - \text{measured}|}{\text{measured}} \times 100\%

Round to one decimal place. [2]

c

Advise the nurse whether this model alone is sufficient for making clinical dosing decisions. In your response, discuss at least two limitations of the model and justify your advice by interpreting your percentage error results in a medical context. [4]

Solutions

36QuestionGraphing 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

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

38QuestionEvaluate compositeAssessment Practice

2 marks~3 minCriterion D

A mobile phone plan charges a fixed monthly fee of 20 dollars plus 0.10 dollars per minute of usage. A tax of 10% is applied to the total bill. The total cost in dollars is modelled by

C(t) = 1.1(20 + 0.10t)

where

t

is the number of minutes used in a month.

a

Explain why multiplying only the fixed fee of 20 dollars by 1.1, rather than the entire expression

(20 + 0.10t)

, produces an understated total cost. [1]

b

Discuss one limitation of using this model to predict a customer's monthly bill. [1]

Solutions

39QuestionPredictionAssessment Practice

2 marks~3 minCriterion D

A ski slope is modelled by the line

y = 2x + 1

, where

x

is the horizontal distance (m) and

y

is the vertical height (m).

Safety regulations state that a ski slope must make an angle of less than 60° with the horizontal to be classified as a beginner slope.

Justify whether this slope qualifies as a beginner slope. [2]

Solutions

40QuestionPredictionAssessment Practice

6 marks~9 minCriterion D

A city planner is modelling the population growth of a town. Census data shows the population was 10 000 in 2000, 12 000 in 2005, 14 400 in 2010, and 17 280 in 2015. Two models are proposed, where

t

is years since 2000:

Model A (exponential):

P(t) = 10000 \times e^{0.0365t}

Model B (logistic):

\displaystyle P(t) = \frac{50000}{1 + 4e^{-0.05t}}

a

Calculate the predicted population in 2030 using both models. Show all working. [2]

b

Justify which model is more appropriate for long-term population prediction, referring to the structure of each model. [2]

c

The city planner must decide whether to commission a new water treatment plant with capacity for 30 000 residents. Advise the city planner whether either model provides sufficient justification to commission the plant for 2030, identifying one assumption of each model that could affect the reliability of this advice. [2]

Solutions

41QuestionReal-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

42QuestionInequalitiesAssessment Practice

6 marks~9 minCriterion B

The diagram shows the feasible region

R

defined by the three inequalities

x \geq 2, \quad y \geq 1, \quad x + y \leq 10,

where

x

and

y

are positive integers.

a

Write down the number of integer-coordinate points that lie on the boundary line

x + y = 10

within region

R

. [1]

b

For each integer value of

x

from

x = 2

to

x = 9

, find the total number of integer-coordinate points in region

R

. [2]

c

The column counts for

x = 2, 3, 4, \ldots, 9

form a sequence. Describe the pattern in this sequence and justify whether the general rule

C(n) = 10 - n

, for

2 \leq n \leq 9

, correctly models the number of valid integer points in column

x = n

. [3]

Solutions

43QuestionInequalitiesAssessment Practice

10 marks~15 minCriterion D

A small business produces two types of wooden toys: chairs (

x

units per week) and tables (

y

units per week). Each chair requires 2 board-feet of wood and 3 hours of labour; each table requires 4 board-feet of wood and 2 hours of labour. Weekly resources are limited to at most 120 board-feet of wood and 90 hours of labour. A contract requires at least 10 chairs per week. Profit is 40 dollars per chair and 60 dollars per table.

a

State the system of inequalities that models this situation. [2]

b

Construct the feasible region and state the coordinates of all corner points. [3]

c

Deduce the production mix that maximises weekly profit, showing your reasoning. [2]

d

Justify whether the optimal solution of 15 chairs and 22.5 tables per week is operationally viable for the business, referring to two assumptions of the linear programming model. [3]

Solutions

44QuestionObjective functionAssessment Practice

6 marks~9 minCriterion C

A factory produces two products, X and Y. Each unit of X earns a profit of 3 dollars and each unit of Y earns a profit of 2 dollars. Daily production is subject to the following constraints, where

x

is the number of units of X and

y

is the number of units of Y:

x + y \leq 8

2x + y \leq 12

x \geq 0, \quad y \geq 0

The profit function is

P = 3x + 2y

(in dollars).

a

Write down the coordinates of the four vertices of the feasible region. [2]

b

Find the maximum daily profit, stating the values of

x

and

y

that achieve it. [2]

c

The factory manager proposes producing 5 units of X and 3 units of Y. Advise the manager whether this production plan should be adopted, justifying your answer with reference to the constraints and the optimal solution. [2]

Solutions

Functions — Free MYP4 Mathematics (Extended) Practice Questions