Quadratic Functions
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Quadratic Functions — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionGraphing parabolasConcept Practice
4 marks~6 minCriterion A
A sports analyst models the height of a basketball shot (in metres) using the function f(x)=−x2+4x+12f(x) = -x^2 + 4x + 12, where xx is the horizontal distance (in metres) from the player.
a
Determine the equation of the axis of symmetry of the parabola. [1]
b
Deduce the coordinates of the vertex of the parabola, showing your method. [2]
c
The basketball hoop is positioned at a height of 15 m and a horizontal distance of 3 m from the player. Justify whether the shot reaches the hoop, using the model to support your answer. [1]
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2QuestionStandard formAssessment Practice
4 marks~6 minCriterion C
A road engineer models the cross-section of a speed bump using the function f(x)=2x2−8x+6f(x) = 2x^2 - 8x + 6, where xx is the horizontal distance in metres from a reference point and f(x)f(x) is the height in metres. The graph of f(x)f(x) is provided.
a
State the coordinates of the yy-intercept of f(x)f(x). [1]
b
Find the xx-intercepts of f(x)f(x) by solving f(x)=0f(x) = 0. [1]
c
A colleague proposes the alternative model g(x)=x2−4x+3g(x) = x^2 - 4x + 3. Justify whether f(x)f(x) or g(x)g(x) produces the steeper speed-bump profile, and advise the engineer which model to select if road safety regulations require the steepest possible cross-section. [2]
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3QuestionAxis 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)=ax2+bx+ch(x) = ax^2 + bx + c, where hh is the height (m) and xx is the horizontal distance (m). The nozzle is at x=0x = 0 and the water lands at x=dx = d. The designer wants to maximise the horizontal distance dd the water travels before hitting the ground.
a
State the axis of symmetry of h(x)=ax2+bx+ch(x) = ax^2 + bx + c and explain what it represents in this context. [2]
b
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 −b2a=d2-\dfrac{b}{2a} = \dfrac{d}{2}. [2]
c
A wall is located at x=3x = 3 m and has height 4 m. The designer proposes the model h(x)=−x2+6xh(x) = -x^2 + 6x. Assess whether the designer's proposal should be accepted. [2]
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4QuestionVertex formAssessment Practice
6 marks~9 minCriterion D
A company models its daily profit PP (in dollars) from selling a product at a price of xx dollars per unit using the quadratic function

P(x)=−2(x−50)2+5000.P(x) = -2(x - 50)^2 + 5000.
a
Interpret the vertex of P(x)P(x) to state the maximum daily profit and the price per unit at which it occurs. [1]
b
The break-even points occur when P(x)=0P(x) = 0. Solve

−2(x−50)2+5000=0-2(x - 50)^2 + 5000 = 0

to find the two prices at which the company breaks even. [2]
c
The company is considering setting the price at either 20 dollars or 80 dollars per unit. Calculate P(20)P(20) and P(80)P(80), then advise the company which price to set, justifying your recommendation with reference to both the mathematical result and one limitation of the model. [3]
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