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

1QuestionMidpoint and Distance Between Two PointsConcept Practice

2 marks~3 minCriterion A

A city planner is designing a straight pedestrian path between two landmarks. On a coordinate grid where each unit represents 100 metres, landmark

A

is located at

(2, 1)

and landmark

B

is located at

(6, 5)

.

A rest shelter must be placed exactly halfway along the path.

Using the midpoint formula

M = \left( \frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2} \right),

calculate the coordinates of the midpoint

M

of segment

AB

, and hence interpret the real-world position of the shelter relative to landmark

A

, in metres, along each axis direction. [2]

Solutions

2QuestionReal-Life Uses of Cartesian CoordinatesConcept Practice

2 marks~3 minCriterion B

A taxi starts at the origin of a coordinate grid, where the horizontal axis represents time

t

(minutes) and the vertical axis represents distance

d

(kilometres). The following positions are recorded:

t = 0

:

(0,\ 0)

\quad

t = 1

:

(1,\ 2)

\quad

t = 2

:

(2,\ 4)

\quad

t = 3

:

(3,\ 6)

Describe the relationship between

t

and

d

using the general ordered-pair notation

(t,\ f(t))

, and interpret what this relationship means for the taxi's journey. [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

4QuestionParallel 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

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

6QuestionGraphing 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

7QuestionSketching y = ax2 + bx + cConcept Practice

4 marks~6 minCriterion B

Consider the three quadratic functions below, along with their corresponding graphs:

$y = x^2 - 4x + 1$ (Graph A)

y = x^2 - 4x + 4

(Graph B)
*

y = x^2 - 4x - 2

(Graph C)

a

Investigate the relationship between the constant term in each equation and the

y

-intercept of its graph. Describe the pattern you observe.

b

Investigate the relationship between the constant term in each equation and the

y

-coordinate of the vertex of the parabola. Describe the pattern you observe.

c

Generalize a rule for how changing the constant term

c

in the quadratic function

y = ax^2 + bx + c

affects the vertical position of the graph.

Solutions

8QuestionGradient 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

9QuestionGradient 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

10QuestionStretching 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

11QuestionEstimating Values and Trends from GraphsAssessment Practice

4 marks~6 minCriterion A

A car journey is represented by a distance–time graph with three straight-line segments.

Segment 1 (

0

h to

2

h): speed

= 30

km/h
Segment 2 (

2

h to

3.5

h): speed

= 60

km/h
Segment 3 (

3.5

h to

5

h): speed

= 45

km/h

The car starts at

d = 0

km.

a

Construct a table of values showing the distance at

0.5

-hour intervals from

t = 0

h to

t = 5

h. [2]

b

Deduce a linear equation of the form

d = mt + c

for the distance during Segment 2. [1]

c

The driver claims the car reached

100

km before

t = 2.7

h. Justify whether this claim is correct. [1]

Solutions

12QuestionEstimating Values and Trends from GraphsAssessment Practice

5 marks~8 minCriterion C

The graph shows the rate of change of a function

f(x)

over

0 \le x \le 8

. The gradient graph is piecewise linear, passing through

(0,\ 4)

,

(2,\ 0)

,

(4,\ -2)

,

(6,\ 0)

, and

(8,\ 3)

. It is given that

f(0) = 5

, and that the area between the gradient graph and the

x

-axis from

x = 0

to any value

x

equals the change in

f(x)

over that interval.

a

Deduce the

x

-coordinates where

f(x)

has stationary points. [1]

b

Justify the classification of each stationary point as a local maximum or local minimum, using the sign of the gradient on either side. [2]

c

A designer claims that

f(x)

never falls below its starting value on

0 \le x \le 8

. Assess whether this claim is correct, supporting your answer with estimated values of

f(x)

at the stationary points. [2]

Solutions

13QuestionReading Distance-Time GraphsAssessment Practice

6 marks~9 minCriterion D

A distance–time graph shows a delivery journey from point

A

to point

D

with three segments:

- Segment

AB

:

A(0,\ 0)

to

B(2,\ 60)

- Segment

BC

:

B(2,\ 60)

to

C(5,\ 60)

- Segment

CD

:

C(5,\ 60)

to

D(t,\ 0)

, completed in

3

hours

Distance is in kilometres; time is in hours.

a

Calculate the gradient of segment

AB

and the coordinates of point

D

. [2]

b

Deduce the gradient of segment

CD

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

c

A logistics company requires the return speed to be at least

\dfrac{2}{3}

of the outbound speed. Evaluate whether this journey meets the requirement and justify your advice to the company. [2]

Solutions

14QuestionIdentifying 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

15QuestionParallel and Perpendicular Lines IntroductoryAssessment Practice

4 marks~6 minCriterion B

An urban planner is designing a pedestrian grid. Two straight paths must meet at a perfect right angle to satisfy accessibility regulations.

The four candidate path pairs are:

Pair 1:

y = 2x + 1

and

y = -\dfrac{1}{2}x + 3

Pair 2:

y = 3x - 2

and

y = -\dfrac{1}{3}x + 5

Pair 3:

y = x + 4

and

y = -x - 1

Pair 4:

y = 4x

and

y = -\dfrac{1}{4}x + 2

a

For each pair, calculate the product of the two slopes. State what you observe. [1]

b

The angle

\theta

between two lines with slopes

m_1

and

m_2

is given by

\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|

Apply this formula to Pairs 1 and 3. Deduce the value of

\theta

for each, and explain why the formula behaves as it does for these two pairs. [2]

c

The accessibility regulation requires paths to meet at exactly 90°. Justify whether all four pairs satisfy this regulation, and state the general algebraic condition that guarantees perpendicularity. [1]

Solutions

16QuestionIdentifying Slope and Y-Intercept AI Generate Add Questions SummaryAssessment Practice

4 marks~6 minCriterion D

A delivery drone's battery charge

C

(percent) decreases linearly with horizontal distance

d

(km) from its base. At

d = 1

km the charge reads

2.5\%

; at

d = 4

km the charge reads

1.25\%

.

a

Calculate the slope of the line passing through

(1,\ 2.5)

and

(4,\ 1.25)

. [1]

b

Determine the equation of the line in the form

C = md + b

. [2]

c

Interpret what

b

represents in this context, and justify whether this model should be used to predict battery charge over a long-distance flight. [1]

Solutions

17QuestionSketching y = ax2 + bx + cAssessment Practice

6 marks~9 minCriterion C

The graph below shows a quadratic function. The vertex is at (2, -1) and the y-intercept is at (0, 3). The x-intercepts are at (1, 0) and (3, 0). Explain, step-by-step, how to determine the equation of the quadratic function in the form

y = ax^2 + bx + c

from the information given in the graph. Justify each step, showing your reasoning clearly using algebraic notation.

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

19QuestionUsing Slope Relationships in GraphsAssessment Practice

4 marks~6 minCriterion B

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

20QuestionVerifying Line Relationships AlgebraicallyAssessment Practice

4 marks~6 minCriterion D

A landscape designer is planning a sprinkler system for a rectangular garden. A main water pipe runs along the line

y = 2x + 1

where units are in metres. A perpendicular sprinkler line must pass through the point

(3, 5)

, where a new flower bed is located.

a

Deduce the equation of the sprinkler line in slope-intercept form. [2]

b

The designer assumes the main pipe has an exact slope of

2

. Advise the designer whether this linear model is sufficient for final installation of the sprinkler line, given that the measured slope of the main pipe may vary within

2 \pm 0.1

and that physical obstacles may exist. [2]

Solutions

21QuestionTranslating Graphs Vertically and HorizontallyAssessment Practice

6 marks~9 minCriterion B

A drone delivery company models the height of a package above ground during a test drop using the parent function

f(x) = x^2

, where

x

is horizontal distance (metres) from the drop point. Engineers test three adjusted flight paths:

g(x) = x^2 + 1

,

h(x) = x^2 - 3

, and

j(x) = x^2 + 5

.

a

Calculate the

y

-values for

f(x)

,

g(x)

,

h(x)

, and

j(x)

at

x = -2,\ -1,\ 0,\ 1,\ 2

. Complete the table below. [2]

x

:

-2

,

-1

,

0

,

1

,

2

f(x)

: \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_

g(x)

: \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_

h(x)

: \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_

j(x)

: \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_

b

Analyse the relationship between the

y

-values of

g(x)

,

h(x)

, and

j(x)

and those of

f(x)

. Deduce a general rule describing how adding a constant

c

to

x^2

transforms the graph of

y = x^2

. [2]

c

A safe landing requires the package height at

x = 0

to remain above

-5

metres. A new flight path is modelled by

k(x) = x^2 - 7

. Using your rule from part (b), advise the engineers whether this flight path should be approved, justifying your answer with reference to the safety requirement. [2]

Solutions

22QuestionCombining 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

23QuestionCombining 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

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