Graphs and Relations — Free IB MYP4 Mathematics (Extended) Questions

1QuestionStretching and Compressing GraphsConcept Practice

4 marks~6 minCriterion A

A solar panel manufacturer models the power output (in watts) of a panel using the function

f(x) = x^2

, where

x

is the number of hours after sunrise. A premium panel has greater output, modelled by

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

.

The premium panel produces 48 watts exactly 4 hours after sunrise.

a

Deduce the value of

a

. [2]

b

Explain what the value of

a

means for the power output of the premium panel compared to the standard panel at every hour of the day. [2]

Solutions

2QuestionConnecting Graphs to Real SituationsConcept Practice

2 marks~3 minCriterion D

A delivery driver completes a 60 km urban route. A distance–time graph of the journey shows three distinct sections: a steep rise, a horizontal segment, and a gentler rise.

a

Interpret what each section of the graph reveals about the driver's speed during the journey. [1]

b

The driver's average speed for the entire journey is calculated as

\dfrac{\text{total distance}}{\text{total time}}

. Advise the logistics manager whether this value is sufficient to plan reliable delivery schedules for similar future routes. [1]

Solutions

3QuestionConnecting Graphs to Real SituationsConcept Practice

4 marks~6 minCriterion A

A car journey is shown on the distance–time graph.

- Segment A:

(0, 0)

to

(2, 120)

- Segment B:

(2, 120)

to

(5, 120)

- Segment C:

(5, 120)

to

(7, 0)

Time is in hours; distance is in kilometres.

a

Calculate the speed of the car during Segment A. [1]

b

Calculate the total distance travelled during the entire journey. [1]

c

The legal speed limit on this road is 55 km/h. Advise the driver whether they broke the speed limit at any point during the journey, justifying your answer with the speeds from all three segments. [2]

Solutions

4QuestionPlotting Points in Four QuadrantsConcept Practice

2 marks~3 minCriterion A

A drone delivery company maps its service area using a Cartesian coordinate system. The depot is at the origin. A customer's location is recorded as the point

(3, -2)

.

a

Construct a Cartesian grid showing all four quadrants and plot the point

(3, -2)

, clearly labelling the point and all four quadrants. [1]

b

The company offers same-day delivery only to customers in Quadrant IV. Justify whether this customer qualifies for same-day delivery. [1]

Solutions

5QuestionInterpreting Grids and Map CoordinatesConcept Practice

2 marks~3 minCriterion B

A city planner records the relationship between the number of bus stops,

x

, and the total number of passengers served per hour,

y

, at four locations:

A(1,\ 2)

,

B(2,\ 4)

,

C(3,\ 6)

,

D(4,\ 8)

a

Deduce the equation that models the relationship between

x

and

y

. [1]

b

The planner states: "Adding a fifth bus stop will serve at least 12 passengers per hour." Justify whether this claim is correct, using your equation. [1]

Solutions

6QuestionInterpreting Grids and Map CoordinatesConcept Practice

2 marks~3 minCriterion C

A search-and-rescue team uses a topographic map with a 4-figure grid reference system to locate a missing hiker. The map grid divides the terrain into squares, each representing a 1 km × 1 km area on the ground.

Explain how a 4-figure grid reference identifies a position on the map, and describe one limitation of this system for pinpointing the hiker's exact location. [2]

Solutions

7QuestionParallel and Perpendicular Lines IntroductoryConcept Practice

2 marks~3 minCriterion A

A city planner is designing a rectangular street grid. On a map, one main street passes through points

A(1, 2)

and

B(5, 10)

.

a

Calculate the slope of the main street. [1]

b

A proposed avenue has equation

y = 2x + 3

. Justify whether this avenue should be approved as a street running parallel to the main street. [1]

Solutions

8QuestionGraphing Simple Cubic FunctionsConcept Practice

4 marks~6 minCriterion D

A water storage tank is designed so that its volume

V

(in litres) relates to a single length measurement

x

(in metres) by

V = 1000x^3

.

The graph of

y = x^3

is shown below.

a

State the coordinates of any intercepts of

y = x^3

and describe its end behaviour using the notation

x \to \pm\infty

. [2]

b

The tank must hold at least 8000 litres. Using the graph of

y = x^3

, justify whether

x = 2

metres meets this design requirement. [2]

Solutions

9QuestionGraphing Simple Cubic FunctionsConcept Practice

2 marks~3 minCriterion A

An engineer models the vibration of a bridge cable using the cubic function

y = x^3 - 2x

, where

x

represents time (seconds) and

y

represents displacement (metres) from equilibrium. The graph of this function is shown below.

State the number of

x

-intercepts of the graph and the coordinates of the

y

-intercept. [2]

Solutions

10QuestionGradient of Parallel LinesConcept Practice

2 marks~3 minCriterion A

A city planner is designing two parallel cycle lanes on a coordinate grid, where each unit represents 50 metres. Lane A passes through the points

(0, 1)

and

(3, 3)

.

(a) Deduce the gradient of Lane B. [2]

Solutions

11QuestionStretching and Compressing GraphsAssessment Practice

2 marks~3 minCriterion B

A camera lens models image distortion using

f(x) = x^2

. Two adjusted lens settings produce

g(x) = (2x)^2

and

h(x) = (0.5x)^2

.

Selected coordinate pairs giving equal output values:

f(x)

:

x

values

1, 2, 3

;

\quad f(x)

values

1, 4, 9

g(x)

:

x

values

0.5, 1, 1.5

;

\quad g(x)

values

1, 4, 9

h(x)

:

x

values

2, 4, 6

;

\quad h(x)

values

1, 4, 9

Deduce the general rule for the horizontal transformation of

y = f(x)

when written as

y = f(kx)

, where

k > 0

, by identifying the pattern in the

x

-coordinates across all three functions. [2]

Solutions

12QuestionCombining Multiple TransformationsAssessment Practice

4 marks~6 minCriterion D

A telecommunications company models the height (in km) of a satellite dish's signal beam using

y = -2(x-3)^2 + 5

where

x

is the horizontal distance in kilometres from the dish. The parent function is

y = x^2

.

a

Describe the sequence of transformations applied to

y = x^2

to obtain

y = -2(x-3)^2 + 5

. [2]

b

Explain one mathematical advantage of using this form of the equation to identify the optimal signal coverage position. [1]

c

The company claims the model is reliable for all distances

x \geq 0

. Advise the company whether this claim should be accepted, supporting your answer with both a mathematical and a physical reason. [1]

Solutions

13QuestionDescribing Graph Changes AlgebraicallyAssessment Practice

7 marks~11 minCriterion C

The diagram shows part of the graph of

y = f(x)

, which passes through the points

A(-2,\, 5)

,

B(0,\, -3)

, and

C(4,\, 1)

.

The table below shows the coordinates of

A

,

B

,

C

and their images under three transformations.

Original on

y = f(x)

:

(-2,\ 5)

\quad

(0,\ -3)

\quad

(4,\ 1)

Image on

y = f(x + 3)

:

(-5,\ 5)

\quad

(-3,\ -3)

\quad

(1,\ 1)

Image on

y = f(x - 2)

:

(0,\ 5)

\quad

(2,\ -3)

\quad

(6,\ 1)

Image on

y = f(x + 5)

:

(-7,\ 5)

\quad

(-5,\ -3)

\quad

(-1,\ 1)

a

State the change in the

x

-coordinate of each point when

y = f(x)

is transformed to

y = f(x + 3)

, and state the direction of the resulting shift. [1]

b

A student claims that the combined transformation

y = f\bigl((x - 2) + 5\bigr)

is equivalent to a single horizontal shift of

y = f(x)

. Find the image of

C(4,\, 1)

under this combined transformation, and hence determine the single equivalent transformation, giving its equation in the form

y = f(x + a)

. [3]

c

Using the pattern in the table, write a general rule for the horizontal shift produced by

y = f(x + a)

for any real constant

a

. Hence justify why

y = f(x - 5)

shifts the graph of

y = f(x)

exactly

5

units to the right, referring explicitly to at least two rows of the table to support your reasoning. [3]

Solutions

14QuestionConnecting Graphs to Real SituationsAssessment Practice

4 marks~6 minCriterion C

A cyclist rides a hilly route. Her distance from the start is recorded every hour:

Time (h):

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

Distance (km):

0,\ 12,\ 20,\ 24,\ 30

A second cyclist rides a flat road at constant speed:

Time (h):

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

Distance (km):

0,\ 15,\ 30,\ 45,\ 60

a

Construct a distance–time graph showing both cyclists on the same axes. [1]

b

Explain why one graph is a straight line and the other is a curve, referring to each cyclist's speed. [2]

c

The hilly-route cyclist claims she maintained an "acceptable average speed" of at least

8

km/h throughout every individual hour of her journey. Justify whether this claim is valid. [1]

Solutions

15QuestionEstimating Values and Trends from GraphsAssessment Practice

4 marks~6 minCriterion D

A car travels away from a city. At

t = 0

, the car is 20 km from the city. The graph shows the car's distance from the city over 6 hours. The relationship is linear.

a

Use the graph to estimate the distance of the car from the city at

t = 2.5

hours. [1]

b

Use the graph to estimate the time at which the car is 120 km from the city. [1]

c

The speed limit on this road is 30 km/h. Using the formula

\text{average speed} = \frac{\text{distance travelled}}{\text{time taken}}

calculate the average speed of the car between

t = 1

h and

t = 5

h, then advise a traffic officer whether the car should be flagged for exceeding the speed limit during this interval. [2]

Solutions

16QuestionIdentifying Coordinates from a GraphAssessment Practice

4 marks~6 minCriterion D

A city planner records the cumulative length of a new cycle path at three checkpoints:

A(1,\ 2)

,

B(3,\ 6)

,

C(5,\ 10)

where the

x

-axis shows time elapsed (hours) and the

y

-axis shows distance covered (kilometres). The three points are plotted on the provided graph.

a

Calculate the slope of the line passing through points

A

and

B

. [1]

b

Explain why point

C

lies on the same straight line as

A

and

B

. [1]

c

Interpret what the slope represents in this context, then advise the city planner whether the current construction rate is sufficient to complete the 20 km path within 9 hours, justifying your answer with a calculation. [2]

Solutions

17QuestionIdentifying Slope and Y-Intercept AI Generate Add Questions SummaryAssessment Practice

6 marks~9 minCriterion B

A city planner models road gradients using equations of the form

ax + by = c

, where

x

is horizontal distance and

y

is elevation. Four road segments are recorded:

Segment 1:

2x + 3y = 6

, giving

y = -\dfrac{2}{3}x + 2

Segment 2:

4x - 2y = 8

, giving

y = 2x - 4

Segment 3:

-3x + 6y = 12

, giving

y = \dfrac{1}{2}x + 2

Segment 4:

5x + 5y = 15

, giving

y = -x + 3

a

Analyse the four segments to describe the relationship between the coefficients

a

,

b

,

c

and the slope

m

and

y

-intercept

d

in

y = mx + d

. [2]

b

Deduce a general rule expressing

m

and

d

in terms of

a

,

b

, and

c

. [1]

c

A proposed road follows

7x - 4y = 20

. Apply your rule to determine the slope and

y

-intercept, then verify by converting the equation to the form

y = mx + d

. [2]

d

A second road follows

2x + 8y = -16

. The planner states that any road with a slope steeper than

-\dfrac{1}{3}

is unsafe for heavy vehicles. Advise the planner whether this road should be approved for heavy-vehicle use, justifying your answer with a numerical comparison. [1]

Solutions

18QuestionDrawing Graphs from y = mx + cAssessment Practice

2 marks~3 minCriterion C

A city's water authority monitors daily water usage. A linear model shows that usage increases at a constant rate as population grows. The line passes through

(0,\ 2)

and

(3,\ 5)

, with population (in thousands) on the

x

-axis and daily water usage (in megalitres) on the

y

-axis.

a

Deduce the gradient

m

and

y

-intercept

c

, and write the equation of the line in the form

y = mx + c

. [1]

b

The authority warns that usage exceeding 6 megalitres per day risks supply shortages. Justify whether a population of 4000 triggers this warning. [1]

Solutions

19QuestionDrawing Graphs from y = mx + cAssessment Practice

2 marks~3 minCriterion D

A mobile phone plan charges a fixed monthly fee of 10 dollars plus 0.05 dollars per megabyte (MB) of data used. The total monthly cost

C

(in dollars) for

d

MB of data is modelled by

C = 0.05d + 10, \quad 0 \leq d \leq 200.

a

Construct a graph of

C

against

d

for the given domain. Label both axes and indicate the

C

-intercept. [1]

b

Critique the linear model's suitability for representing the actual cost of a mobile phone plan, identifying one specific feature of real plans that the model fails to capture. [1]

Solutions

20QuestionGraphing Simple Cubic FunctionsAssessment Practice

6 marks~9 minCriterion B

A packaging engineer models container volume using

y = x^3

, where

x

is the side length in centimetres and

y

is the volume in cm³. A second design scales volume by a factor of 2, modelled by

y = 2x^3

.

x

:

-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3

y = x^3

:

-27,\ -8,\ -1,\ 0,\ 1,\ 8,\ 27

y = 2x^3

:

-54,\ -16,\ -2,\ 0,\ 2,\ 16,\ 54

a

Explain how the

y

-values of

y = 2x^3

relate to those of

y = x^3

, identifying the pattern shown in the table. [2]

b

Deduce the

y

-values of

y = 3x^3

at

x = -2

and

x = 4

, using the pattern from part (a) rather than direct calculation. [2]

c

A third design requires a volume output of at least 150 cm³ when

x = 4

. The engineer proposes

y = ax^3

with

a = 2

. Advise the engineer whether

a = 2

is sufficient and state the minimum integer value of

a

that meets the requirement. [2]

Solutions

21QuestionGraphing and Identifying Exponential Growth and DecayAssessment Practice

4 marks~6 minCriterion D

A wildlife reserve monitors a rare beetle species over four consecutive years. The population is modelled by an exponential function. The graph shows population

P

(in hundreds) against time

t

(in years), with points

(0, 3)

,

(1, 6)

,

(2, 12)

, and

(3, 24)

plotted and labelled.

a

State the initial population of beetles and deduce the growth factor. [2]

b

Write the equation of the exponential model in the form

P = a \cdot b^t

. [1]

c

The reserve can sustainably support a maximum of 5000 beetles. Justify whether the reserve manager must intervene before year

t = 5

. [1]

Solutions

22QuestionVerifying Line Relationships AlgebraicallyAssessment Practice

6 marks~9 minCriterion B

An urban planner is designing a city block. Two roads are being built: Road A has slope

m_1 = \dfrac{3}{4}

and Road B has slope

m_2 = -\dfrac{4}{3}

. A third road, Road C, runs parallel to Road A with slope

m_3 = \dfrac{3}{4}

.

a

Calculate

m_1 \times m_2

and

m_1 \times m_3

. [2]

b

Deduce the general algebraic relationship between the slopes of two perpendicular lines and between the slopes of two parallel lines. [2]

c

The planner claims Roads A and B form a perfect right-angle intersection, required by the city's safety code. Justify whether this claim is correct, and advise the planner on what adjustment to Road B's slope would be needed if it were instead built with slope

m = -\dfrac{5}{4}

. [2]

Solutions

23QuestionUsing Slope Relationships in GraphsAssessment Practice

4 marks~6 minCriterion C

A city architect is designing two straight pedestrian pathways that must meet at a perfect right angle to ensure safe sightlines at an intersection.

The architect models the pathways on a coordinate grid.

Pathway A:

y = \dfrac{3}{4}x - 2

Pathway B:

y = -\dfrac{4}{3}x + 5

a

Calculate the slope of each pathway. [1]

b

Deduce, using your results from part (a), the general rule relating the slopes of any two perpendicular lines. [1]

c

The architect proposes a third pathway,

y = \dfrac{2}{5}x + 1

, to intersect Pathway A at a right angle. Advise the architect whether this proposed pathway satisfies the perpendicularity condition, and state the correction required if it does not. [2]

Solutions

24QuestionGradient of Perpendicular LinesAssessment Practice

2 marks~3 minCriterion D

A city engineer is designing two straight roads that must intersect at a right angle. Road

L_1

has a gradient of

2

.

a

Deduce the gradient of road

L_2

, given that

L_1 \perp L_2

. [1]

b

The engineer states: "A road with a gradient steeper than

-1

is unsafe for heavy vehicles." Advise the engineer whether road

L_2

is suitable for heavy vehicles, justifying your answer with an inequality comparison. [1]

Solutions

Graphs and Relations — Free MYP4 Mathematics (Extended) Practice Questions