Graphs and Relations — Free IB MYP5 Mathematics (Standard) Questions

1QuestionConnecting Graphs to Real SituationsConcept Practice

2 marks~3 minCriterion A

A cyclist's journey is shown on the distance–time graph.

Segment A:

t = 0

to

t = 10

s, distance increases from

0

m to

20

m.
Segment B:

t = 10

s to

t = 20

s, distance remains constant at

20

m.
Segment C:

t = 20

s to

t = 30

s, distance decreases from

20

m to

0

m.

a

Calculate the gradient of Segment A. [1]

b

Interpret what the gradient of Segment B tells you about the cyclist's motion during that interval. [1]

Solutions

2QuestionReading Distance-Time GraphsConcept Practice

3 marks~5 minCriterion C

A cyclist travels from town A to town B. The distance–time graph for the journey contains a horizontal segment.

a

Explain what the horizontal segment indicates about the cyclist's motion. [1]

b

Discuss one limitation of using a distance–time graph to represent this journey. In your response, refer to specific information that the graph does not show. [2]

Solutions

3QuestionInterpreting 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

4QuestionIdentifying Coordinates from a GraphConcept Practice

1 mark~2 minCriterion A

Identify the coordinates of point P on the Cartesian plane.

a

(2, -3)

b

(-3, 2)

c

(3, -2)

d

(-2, 3)

Solutions

5QuestionParallel and Perpendicular Lines IntroductoryConcept Practice

2 marks~3 minCriterion C

A city planner is designing two straight roads on a coordinate grid. Road A follows the equation

y = 2x + 1

.

Road B must be built perpendicular to Road A, passing through the point

(4, -1)

.

a

Deduce the slope of Road B. [1]

b

The planner claims Road B can be written as

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

. Justify whether this equation correctly represents Road B. [1]

Solutions

6QuestionDrawing Graphs from y = mx + cConcept Practice

2 marks~3 minCriterion A

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

7QuestionSketching y = ax2 + bx + cConcept Practice

4 marks~6 minCriterion B

The graphs of four quadratic functions are shown below. All graphs share the same vertex at (0, 0).

Graph 1:

y = x^2

Graph 2:

y = 3x^2

Graph 3:

y = 0.5x^2

Graph 4:

y = -x^2

Graph	Equation	Opens	Width Compared to $y=x^2$
1	$y = x^2$	Up	Same
2	$y = 3x^2$	Up	Narrower
3	$y = 0.5x^2$	Up	Wider
4	$y = -x^2$	Down	Same

(a) Investigate the relationship between the coefficient of the

x^2

term (the value of 'a' in

y=ax^2

) and the following characteristics of the parabola:
(i) Whether the parabola opens upwards or downwards.
(ii) The width of the parabola compared to

y=x^2

.
(b) Generalize your findings into rules describing how the value of 'a' affects the graph of the quadratic function.
(c) Predict what the graph of

y = -0.5x^2

would look like. How would its direction and width compare to the other graphs?

Solutions

8QuestionGraphing Simple Cubic FunctionsConcept Practice

2 marks~3 minCriterion A

An engineer models the vertical displacement of a suspension bridge cable using the cubic function

y = x^3 - 4x

, where

x

is the horizontal distance (in metres) from the centre of the bridge and

y

is the vertical displacement (in metres) from the reference level.

The graph of

y = x^3 - 4x

is provided.

a

Deduce the

y

-intercept of the graph, showing your working. [1]

b

Interpret what the

y

-intercept means in the context of the bridge model. [1]

Solutions

9QuestionGradient of Perpendicular LinesConcept Practice

2 marks~3 minCriterion C

An urban planner is designing a city block. Two roads must be built so that they intersect at exactly 90°.

Road A follows the line

y = 3x - 5

.

Road B passes through the point

(6, 1)

and must be perpendicular to Road A.

a

State the gradient of Road B. [1]

b

The planner claims Road B can be represented by

y = -\dfrac{1}{3}x + 3

. Justify whether this equation is correct and whether the road design meets the 90° requirement. [1]

Solutions

10QuestionGradient of Parallel LinesConcept Practice

2 marks~3 minCriterion A

A city planner is designing two parallel roads on a grid map. Road A passes through the points

(2, 3)

and

(6, 11)

.

a

Calculate the gradient of Road A. [1]

b

Road B is parallel to Road A and passes through the point

(0, 5)

. The planner claims Road B will intersect the

y

-axis at a height of 5 map units, representing 50 000 cm in the real world (scale

1 : 10\,000

). Justify whether this claim is mathematically consistent. [1]

Solutions

11QuestionReflections Across AxesConcept Practice

2 marks~3 minCriterion A

A graphic designer reflects the letter 'L', with vertices at

(0, 0)

,

(0, 3)

, and

(1, 0)

, across the

y

-axis to create a symmetrical logo.

a

State the coordinates of the reflected vertices. [1]

b

The client requests that the combined logo — original and reflected 'L' together — forms a single, visually balanced shape. Justify whether this reflection achieves that outcome, referring to the position of the shared vertices. [1]

Solutions

12QuestionConnecting 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

13QuestionConnecting Graphs to Real SituationsAssessment Practice

4 marks~6 minCriterion D

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

14QuestionIdentifying Coordinates from a GraphAssessment Practice

6 marks~9 minCriterion D

Two mobile phone plans are advertised as follows:

Plan X: monthly base fee of 20 dollars, plus a charge per minute used.
Plan Y: no base fee, charged only per minute used.

The graph shows monthly cost (

y

, in dollars) against minutes used (

x

). Plan X passes through

(0,\ 20)

and

(100,\ 50)

. Plan Y passes through

(0,\ 0)

and

(150,\ 60)

.

a

Calculate the equation of each plan in the form

y = mx + c

. [2]

b

Deduce the number of minutes at which both plans cost the same. [2]

c

Advise a customer who expects to use 250 minutes per month on which plan to choose, justifying your recommendation with calculated costs. [2]

Solutions

15QuestionReal-Life Uses of Cartesian CoordinatesAssessment Practice

8 marks~12 minCriterion B

A taxi travels along a straight road through a city grid. GPS tracking records the following positions:

Point A	$x = 1$	$y = 3$
Point B	$x = 2$	$y = 5$
Point C	$x = 3$	$y = 7$
Point D	$x = 4$	$y = 9$

a

Describe the pattern in the

y

-coordinates as

x

increases by 1, and state the constant difference. [2]

b

Deduce the coordinates of the next two points, E and F, on the same road. [2]

c

Determine the equation of the road in the form

y = mx + c

. Explain how the constant difference from part (a) determines the value of

m

. [2]

d

The taxi dispatcher claims the road passes through the point

(10, 21)

. Justify whether this claim is correct, and interpret what your answer means about the taxi's route. [2]

Solutions

16QuestionIdentifying 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

17QuestionDrawing Graphs from y = mx + cAssessment Practice

4 marks~6 minCriterion D

A company purchases a delivery van for 40 000 dollars. The van loses 3 500 dollars in value each year. The value of the van after

x

years is modelled by:

V = 40000 - 3500x

a

Calculate the value of the van after 4 years. [1]

b

Construct a graph of

V

against

x

for

0 \leq x \leq 8

. Label both axes with their quantities and units, and indicate a clear scale. [2]

c

The company plans to sell the van once its value falls below 15 000 dollars. Using your graph or the equation, deduce the earliest year in which the van should be sold, then advise the company whether the linear model is a reliable basis for this decision. [1]

Solutions

18QuestionFinding Vertex and Axis of SymmetryAssessment Practice

8 marks~12 minCriterion D

A drone's height above the ground (in metres) is modelled by

h(t) = -5t^2 + 40t + 10

where

t

is the time in seconds after launch.

a

Determine the time at which the drone reaches its maximum height and calculate that maximum height. [2]

b

The drone operates safely only when its height is at least 15 m. Deduce the time interval during which the drone maintains a safe height. [2]

c

A safety regulation states the drone must remain above 15 m for at least 7 seconds. Using your results, advise the drone operator whether this model provides sufficient assurance of compliance with the regulation, and justify your advice by identifying one reason why the real flight may not match the model's prediction. [4]

Solutions

19QuestionGraphing and Identifying Exponential Growth and DecayAssessment Practice

4 marks~6 minCriterion D

A wildlife reserve records the population of 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)

,

(3, 24)

labelled.

a

State the initial population and deduce the growth factor from the graph. [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. Advise the reserve manager whether intervention is required before year

t = 5

, justifying your answer with a calculation. [1]

Solutions

20QuestionVerifying 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

21QuestionGradient of Parallel LinesAssessment Practice

6 marks~9 minCriterion D

A civil engineer checks whether a proposed road segment is parallel to an existing road modelled by

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

, where

x

and

y

are horizontal and vertical distances in kilometres.

Survey data for the proposed segment gives two control points:

First control point:

(1,\ 1.8)

Last control point:

(7,\ 6.0)

a

State the gradient formula and calculate the gradient of the proposed road segment using the two control points. [3]

b

Deduce whether the proposed road segment is parallel to the existing road, justifying your answer by comparing the two gradients. [1]

c

The engineer requires the two roads to be parallel to within a tolerance of

\pm 0.02

. Advise the engineer whether the proposed segment meets this requirement, and suggest one adjustment to the last control point to achieve exact parallelism. [2]

Solutions

22QuestionCombining Multiple TransformationsAssessment Practice

2 marks~3 minCriterion B

A parabolic solar reflector dish is repositioned on a tracking mount. Its cross-section is modelled by

g(x) = (x - h)^2 + k

, where

h

is the horizontal adjustment (in metres) and

k

is the minimum height (in metres) above the base frame.

Four positioning configurations are recorded:

h

:

2,\ 4,\ -1,\ -3

k

:

1,\ 3,\ 5,\ 2

Analyse the relationship between

h

,

k

, and the vertex of

g

across all four configurations, and state the general rule for the vertex coordinates of

g(x) = (x - h)^2 + k

. [2]

Solutions

23QuestionCombining Multiple TransformationsAssessment Practice

4 marks~6 minCriterion C

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 - 5

.

A safety regulation states that the drone must remain at or above

-4

metres relative to the launch reference height.

a

State the coordinates of the vertex of

g(x)

. [1]

b

Explain how the graph of

g(x)

is obtained from the graph of

f(x)

through translations. [2]

c

Evaluate

g(-2)

and justify whether the drone satisfies the safety regulation at

x = -2

. [1]

Solutions

24QuestionStretching and Compressing GraphsAssessment Practice

4 marks~6 minCriterion D

A civil engineer models the vertical sag of a suspension bridge cable using

y = 0.005x^2

where

y

is the vertical sag in metres and

x

is the horizontal distance in metres from the lowest point.

a

When the load increases by 20%, the engineer applies a vertical stretch by a factor of 1.2. Determine the new function. [1]

b

The bridge has a main span of 400 m, so

x

ranges from

-200

to

200

. Calculate the maximum sag predicted by the new model. [1]

c

The engineer claims the quadratic model is sufficient for all design decisions. Critique this claim, referring to at least two real-world factors not captured by the model. [2]

Solutions

Graphs and Relations — Free MYP5 Mathematics (Standard) Practice Questions