Quadratic Functions

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Quadratic Functions — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionGraphing parabolasConcept Practice

4 marks~6 minCriterion A

A structural engineer models the cross-section of a concrete arch using the equation

y = 2x^2 + 8x + 6

, where

x

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

y

is the height (metres) above ground level.

State the

y

-intercept of the parabola and deduce the direction in which it opens, justifying your answer using the equation. [2]

Calculate the discriminant of

2x^2 + 8x + 6 = 0

and deduce the number of points at which the arch meets ground level. [1]

Justify whether this arch design is structurally feasible, given that a valid arch must contact the ground at exactly two distinct points. [1]

Solutions

2QuestionGraphing 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.

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

Calculate the

x

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

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

3QuestionFinding vertexAssessment Practice

4 marks~6 minCriterion D

A structural engineer models the cross-section of a cable support arch using the function

f(x) = 3(x - 1)^2 + k

, where

x

is the horizontal distance in metres from a reference point and

k

is an unknown constant. The vertex of the arch is at the point

(1, -3)

Deduce the value of

k

. [1]

Calculate the

y

-intercept of

f(x)

and interpret what this value represents in the context of the arch. [2]

The engineer considers adjusting the height of the arch by varying

k

. Justify which values of

k

are structurally valid if the arch must cross ground level (

f(x) = 0

) at two distinct points. [1]

Solutions

4QuestionAxis of symmetryAssessment Practice

6 marks~9 minCriterion B

A water fountain projects a jet of water from a nozzle at ground level. The water follows a parabolic path 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

the water travels before hitting the ground.

State the axis of symmetry of

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

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

The designer claims the maximum horizontal distance is achieved when the vertex lies directly above the midpoint of the water's flight. Using the axis of symmetry, show that this requires

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

. [2]

A wall is located at

x = 3

m and has height 4 m. The designer proposes the model

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

. Assess whether the designer's proposal should be accepted. [2]

Solutions

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