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Graphs and Relations

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

1QuestionCombining Multiple TransformationsConcept 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+kg(x) = (x - h)^2 + k, where hh is the horizontal adjustment (in metres) and kk is the minimum height (in metres) above the base frame.

Four positioning configurations are recorded:

hh: 2, 4, −1, −32,\ 4,\ -1,\ -3

kk: 1, 3, 5, 21,\ 3,\ 5,\ 2

Analyse the relationship between hh, kk, and the vertex of gg across all four configurations, and state the general rule for the vertex coordinates of g(x)=(x−h)2+kg(x) = (x - h)^2 + k. [2]

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2QuestionCombining Multiple TransformationsConcept Practice
4 marks~6 minCriterion A
A drone follows a flight path modelled by f(x)=x2f(x) = x^2, where xx is horizontal distance (metres) and f(x)f(x) is height (metres). An adjusted path is modelled by g(x)=(x+3)2−5g(x) = (x+3)^2 - 5.

A safety regulation states that the drone must remain at or above −4-4 metres relative to the launch reference height.
a
State the coordinates of the vertex of g(x)g(x). [1]
b
Explain how the graph of g(x)g(x) is obtained from the graph of f(x)f(x) through translations. [2]
c
Evaluate g(−2)g(-2) and justify whether the drone satisfies the safety regulation at x=−2x = -2. [1]
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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)(0, 0) to (2,120)(2, 120)
- Segment B: (2,120)(2, 120) to (5,120)(5, 120)
- Segment C: (5,120)(5, 120) to (7,0)(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]
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4QuestionConnecting Graphs to Real SituationsConcept Practice
2 marks~3 minCriterion A
A cyclist's journey is shown on the distance–time graph.

Segment A: t=0t = 0 to t=10t = 10 s, distance increases from 00 m to 2020 m.
Segment B: t=10t = 10 s to t=20t = 20 s, distance remains constant at 2020 m.
Segment C: t=20t = 20 s to t=30t = 30 s, distance decreases from 2020 m to 00 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]
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5QuestionInterpreting 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]
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6QuestionInterpreting and Writing Linear Equations from GraphConcept Practice
2 marks~3 minCriterion A
A city's water authority monitors daily water usage. A graph shows a linear relationship between the number of hours since midnight, xx, and the volume of water (in millions of litres) remaining in a reservoir, yy. The line passes through (0, 1)(0,\ 1) and (2, 2)(2,\ 2).

State the slope and yy-intercept of the line, and write the equation of the line in the form y=mx+by = mx + b. [1]

Interpret what the slope and yy-intercept each represent in this context. [1]
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7QuestionGraphing 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=x3−4xy = x^3 - 4x, where xx is the horizontal distance (in metres) from the centre of the bridge and yy is the vertical displacement (in metres) from the reference level.

The graph of y=x3−4xy = x^3 - 4x is provided.
a
Deduce the yy-intercept of the graph, showing your working. [1]
b
Interpret what the yy-intercept means in the context of the bridge model. [1]
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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−5y = 3x - 5.

Road B passes through the point (6,1)(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=−13x+3y = -\dfrac{1}{3}x + 3. Justify whether this equation is correct and whether the road design meets the 90° requirement. [1]

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9QuestionVerifying Line Relationships AlgebraicallyConcept Practice
2 marks~3 minCriterion A
A city planner is checking whether two proposed roads are perpendicular before approving a junction design. Road L1L_1 passes through points A(1,2)A(1, 2) and B(3,8)B(3, 8). Road L2L_2 passes through points C(4,1)C(4, 1) and D(7,2)D(7, 2).
a
Calculate the gradient of L1L_1 and the gradient of L2L_2. [1]
b
Advise the city planner whether the junction design should be approved on the basis of perpendicularity, justifying your answer using the gradients found in part (a). [1]
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10QuestionStretching and Compressing GraphsAssessment Practice
4 marks~6 minCriterion D
A civil engineer models the vertical sag of a suspension bridge cable using

y=0.005x2y = 0.005x^2

where yy is the vertical sag in metres and xx 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 xx ranges from −200-200 to 200200. 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]
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11QuestionDescribing Graph Changes AlgebraicallyAssessment Practice
5 marks~8 minCriterion C
The diagram shows the graphs of f(x)=x2f(x) = x^2 and its image g(x)g(x) after two transformations applied in order: a vertical stretch by scale factor 33, followed by a vertical translation of −4-4 units.
a
Write down the equation of g(x)g(x). [1]
b
Find the coordinates of the vertex of g(x)g(x) and state the range of g(x)g(x). [2]
c
The graph of g(x)g(x) is translated by (p0)\begin{pmatrix} p \\ 0 \end{pmatrix} to give the graph of h(x)h(x), where h(x)=3x2−12x+8h(x) = 3x^2 - 12x + 8. Justify whether p=2p = 2 is the correct value for this translation vector. [2]
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12QuestionConnecting Graphs to Real SituationsAssessment Practice
6 marks~9 minCriterion C
A cyclist's speed vv (m/s) during a 20-second ride is modelled by

v(t)=−0.05t3+1.5t2,0≤t≤20,v(t) = -0.05t^3 + 1.5t^2, \quad 0 \leq t \leq 20,

and passes through A(0, 0)A(0,\,0), B(5, 8)B(5,\,8), C(10, 12)C(10,\,12), D(15, 10)D(15,\,10), and E(20, 0)E(20,\,0).

The derivative is v′(t)=−0.15t2+3tv'(t) = -0.15t^2 + 3t.
a
Calculate the average rate of change of speed between t=5t = 5 s and t=15t = 15 s. [2]
b
By drawing a tangent to the curve at C(10, 12)C(10,\,12), estimate the instantaneous rate of change of speed at t=10t = 10 s. State the coordinates of the two points used on your tangent line. [2]
c
Calculate v′(10)v'(10) and advise the cyclist's coach whether the rider should increase effort, maintain pace, or begin conserving energy at t=10t = 10 s. Justify your advice using the value obtained and the shape of the graph. [2]
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13QuestionConnecting Graphs to Real SituationsAssessment Practice
6 marks~9 minCriterion D
A city is considering extending a bus route. The projected daily ridership (in thousands of passengers) as a function of route length xx km is

R(x)=0.5x+2R(x) = 0.5x + 2

and the daily operating cost (in thousands of dollars) is

C(x)=0.3x+5.C(x) = 0.3x + 5.

Each passenger pays a fare of 2.50 dollars. The current route length is 10 km, and the city plans to extend the route to 22 km.
a
Calculate the current daily profit, where profit = revenue −- operating cost and revenue = fare ×\times ridership. [2]
b
Determine the route length at which daily revenue equals daily operating cost. [2]
c
Advise the city whether the planned extension to 22 km is financially justified. In your response, identify one limitation of the linear model and explain how it could affect the reliability of your advice. [2]
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14QuestionInterpreting Grids and Map CoordinatesAssessment Practice
3 marks~5 minCriterion D
A park map uses a grid where each unit represents 50 metres. Three landmarks are located at the following coordinates:

Fountain: (3,2)(3, 2)
Picnic Area: (7,6)(7, 6)
Playground: (3,9)(3, 9)
a
Calculate the straight-line distance, in metres, between the Fountain and the Playground. [1]
b
Calculate the straight-line distance, in metres, between the Fountain and the Picnic Area. [1]
c
A path connecting all three landmarks must be under 600 metres in total length to qualify for a park accessibility grant. Justify whether this path qualifies for the grant, using your results from (a) and (b). [1]
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15QuestionIdentifying Coordinates from a GraphAssessment Practice
3 marks~5 minCriterion A
A city planner is designing a straight pedestrian path between two landmarks. Landmark PP is located at (2,5)(2, 5) and landmark QQ is at (8,1)(8, 1) on a scaled map, where each unit represents 100 metres.

A water fountain must be installed at the exact midpoint MM of path PQPQ.
a
Calculate the coordinates of midpoint MM. [1]
b
Deduce the straight-line distance from PP to MM, giving your answer in metres. [1]
c
The city planner states: "The fountain is closer to the city centre at (4,4)(4, 4) than either landmark is." Assess whether this statement is correct, supporting your answer with distance calculations. [1]
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16QuestionReal-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 Ax=1x = 1y=3y = 3
Point Bx=2x = 2y=5y = 5
Point Cx=3x = 3y=7y = 7
Point Dx=4x = 4y=9y = 9
a
Describe the pattern in the yy-coordinates as xx 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+cy = mx + c. Explain how the constant difference from part (a) determines the value of mm. [2]
d
The taxi dispatcher claims the road passes through the point (10,21)(10, 21). Justify whether this claim is correct, and interpret what your answer means about the taxi's route. [2]
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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=cax + by = c, where xx is horizontal distance and yy is elevation. Four road segments are recorded:

Segment 1: 2x+3y=62x + 3y = 6, giving y=−23x+2y = -\dfrac{2}{3}x + 2

Segment 2: 4x−2y=84x - 2y = 8, giving y=2x−4y = 2x - 4

Segment 3: −3x+6y=12-3x + 6y = 12, giving y=12x+2y = \dfrac{1}{2}x + 2

Segment 4: 5x+5y=155x + 5y = 15, giving y=−x+3y = -x + 3
a
Analyse the four segments to describe the relationship between the coefficients aa, bb, cc and the slope mm and yy-intercept dd in y=mx+dy = mx + d. [2]
b
Deduce a general rule expressing mm and dd in terms of aa, bb, and cc. [1]
c
A proposed road follows 7x−4y=207x - 4y = 20. Apply your rule to determine the slope and yy-intercept, then verify by converting the equation to the form y=mx+dy = mx + d. [2]
d
A second road follows 2x+8y=−162x + 8y = -16. The planner states that any road with a slope steeper than −13-\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]

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18QuestionTable of Values and Line PlottingAssessment Practice
5 marks~8 minCriterion C
A city planner models the relationship between the number of weeks since a road project began, xx, and the total kilometres of road completed, yy, using the equation y=2x−3y = 2x - 3.

A technician records the following data:

xx (weeks): −1, 0, 1, 2, 3, 4-1,\ 0,\ 1,\ 2,\ 3,\ 4

yy (km completed): −5, −3, −1, 1, 3, 6-5,\ -3,\ -1,\ 1,\ 3,\ 6
a
Plot all six coordinate pairs on a clearly labelled coordinate grid. [1]
b
Justify which coordinate pair does not lie on the line y=2x−3y = 2x - 3 by substituting each xx-value into the equation and comparing the calculated yy-values with the recorded values. Show all working. [2]
c
Advise the city planner whether the recorded data at week 4 should be used for scheduling decisions, giving a mathematical reason for your recommendation. [2]
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19QuestionDrawing 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 xx years is modelled by:

V=40000−3500xV = 40000 - 3500x
a
Calculate the value of the van after 4 years. [1]
b
Construct a graph of VV against xx for 0≤x≤80 \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]
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20QuestionFinding 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)=−5t2+40t+10h(t) = -5t^2 + 40t + 10

where tt 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]
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21QuestionGraphing Simple Cubic FunctionsAssessment Practice
6 marks~9 minCriterion B
A packaging engineer models container volume using y=x3y = x^3, where xx is the side length in centimetres and yy is the volume in cm³. A second design scales volume by a factor of 2, modelled by y=2x3y = 2x^3.

xx: −3, −2, −1, 0, 1, 2, 3-3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3

y=x3y = x^3: −27, −8, −1, 0, 1, 8, 27-27,\ -8,\ -1,\ 0,\ 1,\ 8,\ 27

y=2x3y = 2x^3: −54, −16, −2, 0, 2, 16, 54-54,\ -16,\ -2,\ 0,\ 2,\ 16,\ 54
a
Explain how the yy-values of y=2x3y = 2x^3 relate to those of y=x3y = x^3, identifying the pattern shown in the table. [2]
b
Deduce the yy-values of y=3x3y = 3x^3 at x=−2x = -2 and x=4x = 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=4x = 4. The engineer proposes y=ax3y = ax^3 with a=2a = 2. Advise the engineer whether a=2a = 2 is sufficient and state the minimum integer value of aa that meets the requirement. [2]

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22QuestionGraphing 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 PP (in hundreds) against time tt (in years), with points (0,3)(0, 3), (1,6)(1, 6), (2,12)(2, 12), (3,24)(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⋅btP = 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=5t = 5, justifying your answer with a calculation. [1]
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23QuestionVerifying 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 m1=34m_1 = \dfrac{3}{4} and Road B has slope m2=−43m_2 = -\dfrac{4}{3}. A third road, Road C, runs parallel to Road A with slope m3=34m_3 = \dfrac{3}{4}.
a
Calculate m1×m2m_1 \times m_2 and m1×m3m_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=−54m = -\dfrac{5}{4}. [2]

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24QuestionVerifying Line Relationships AlgebraicallyAssessment Practice
6 marks~9 minCriterion D
A surveyor uses GPS coordinates to map two straight fence lines on a farm. The first fence line passes through A(2,1)A(2, 1) and B(8,5)B(8, 5). The second fence line passes through C(3,6)C(3, 6) and D(7,0)D(7, 0). The farmer claims the two fence lines are perpendicular and plans to install a gate that fits exactly in the corner between them.
a
Calculate the gradient of each fence line. [2]
b
Justify whether the two fence lines are perpendicular. [2]
c
The GPS has a systematic error of ±5\pm 5 m for each coordinate. Advise the farmer whether it is safe to construct a fixed-angle gate without first taking more precise on-site measurements. [2]
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