Grade 11 · MCR3U · Functions

MCR3U Quadratic Functions Practice Test (with Solutions)

Quadratic Functions is where you get fast at parabolas: completing the square, finding zeros and reading the discriminant. The free test has 8 questions, from writing y = x² - 5x in vertex form to solving a linear-quadratic system with exact values. They are short, so time yourself.

Free Quadratic Functions Practice Test

8 questions, exactly as a free account gets them in the first Quadratic Functions test on Martian Lab. Try each one on paper first, then open the answer.

Express $y = x^2 - 5x$ in vertex form
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$y = \left(x - \dfrac{5}{2}\right)^2 - \dfrac{25}{4}$
What is the equation of the axis of symmetry of the function $y = 2x^2 - 7x - 24$
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$x = \dfrac{7}{4}$
What are the zeros of the function, $y = x^2 + 3x$
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$x = 0,\,-3$
Describe the nature of the roots for the following quadratic equation: $8x^2 - 24x + 18 = 0$
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2 real equal roots
Describe the nature of the roots for the following quadratic equation: $(x - 3)^2 + 2x = 0$
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no real roots
State the equation for a family of parabolas with vertex $(-3, 9)$ in vertex form
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$y = a\,(x+3)^2 + 9,\; a \neq 0$
Solve the linear-quadratic system. Express solution(s) in exact value. $f(x) = 3x^2 + 17x + 16$ $g(x) = -x - 2$
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$3x^2 + 17x + 16 = -x - 2$ $3x^2 + 18x + 18 = 0$ $x^2 + 6x + 6 = 0$ $x = \frac{-6 \pm \sqrt{36 - 24}}{2}$ $x = -3 \pm \sqrt{3}$ $x = -3 + \sqrt{3}, \quad y = 1 - \sqrt{3}$ $x = -3 - \sqrt{3}, \quad y = 1 + \sqrt{3}$ $\therefore$ Points of intersections are $(-3+\sqrt{3},\; 1-\sqrt{3})$ and $(-3-\sqrt{3},\; 1+\sqrt{3})$
Solve the following equation. You cannot use the quadratic formula. $x^2 + \dfrac{8}{3} = \dfrac{10}{3}x$
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$3x^2 + 8 = 10x$ $3x^2 - 10x + 8 = 0$ $(3x - 4)(x - 2) = 0$ $x = \dfrac{4}{3}$ or $x = 2$

This Unit on Martian Lab

The lessons go from an introduction to quadratics, maximums and minimums and problem solving to the quadratic formula and the nature of roots, families of quadratic functions and linear-quadratic systems.

Introduction to Quadratics
Maximum & Minimum of Quadratic Functions
Problem Solving with Quadratics
Quadratic Formula: Nature of Roots
Family of Quadratic Functions
Linear-Quadratic Systems

Then 4 full Quadratic Functions practice tests to check yourself.

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Quadratic Functions FAQ

What is on an MCR3U quadratic functions test?+

Vertex form and completing the square, maximum and minimum problems, zeros, the quadratic formula and the nature of the roots, families of parabolas, and linear-quadratic systems. The free questions on this page cover all of these except the max and min word problems.

How do you tell the nature of the roots?+

Work out the discriminant, b² - 4ac. Positive means two real roots, zero means two equal real roots and negative means no real roots. For 8x² - 24x + 18 = 0 it is 576 - 576 = 0, so the roots are equal.

How do you find the axis of symmetry?+

Use x = -b/(2a). For y = 2x² - 7x - 24 that is x = 7/4. In vertex form, y = a(x - h)² + k, it is just x = h.

What is a family of parabolas?+

All the parabolas that share something, like a vertex or a pair of zeros, written with a free a. The family with vertex (-3, 9) is y = a(x + 3)² + 9, where a ≠ 0.

How do you solve a linear-quadratic system?+

Set the two expressions equal, move everything to one side and solve the quadratic. Substitute each x into the linear equation to get y. The discriminant of that quadratic tells you whether the line crosses the parabola twice, touches it once or misses it.

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