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Quizzes > High School Quizzes > Mathematics

Quadratic Functions and Equations Practice Quiz

Ace your unit test with clear answers

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art depicting a trivia quiz on Quadratic Mastery for high school students.

This quadratic functions quiz helps you practice graphs, forms like standard and vertex, roots, equations, and word problems. You get clear answers after each of the 20 questions. Use it to spot gaps before a test, build speed, and feel ready in class.

What is the standard form of a quadratic equation?
ax^2 + c = 0
x^2 + bx
bx + c = 0
ax^2 + bx + c = 0
The standard form of a quadratic equation is written as ax^2 + bx + c = 0, where a ≠ 0. This form helps in identifying the coefficients and applying solution methods effectively.
For the quadratic function f(x) = ax^2 + bx + c, which graph feature indicates the function's maximum or minimum value?
Y-intercept
X-intercepts
Vertex
Axis of symmetry
The vertex of the parabola is the point where the quadratic function reaches its highest or lowest value. While the axis of symmetry divides the parabola into mirror images, the vertex actually represents the extreme value.
Which of the following methods is NOT typically used to solve quadratic equations?
Completing the square
Substitution method
Quadratic formula
Factorization
Factorization, completing the square, and the quadratic formula are the most common methods for solving quadratic equations. The substitution method is not a standard technique for solving quadratics.
In the quadratic formula x = (-b ± √(b² - 4ac)) / (2a), what does the expression under the square root represent?
Vertex
Discriminant
Axis of symmetry
Coefficient
The expression b² - 4ac is known as the discriminant and it determines the nature of the quadratic equation's roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real repeated root, and a negative discriminant indicates two complex roots.
What is the y-intercept of a quadratic function f(x) = ax^2 + bx + c?
0
b
a
c
The y-intercept of a quadratic function is found by setting x = 0, which yields f(0) = c. Therefore, the constant term c represents the y-intercept.
Given the quadratic equation 2x² - 4x - 6 = 0, what is the value of the discriminant?
16
64
32
48
The discriminant is computed as b² - 4ac. For the equation 2x² - 4x - 6 = 0, we have (-4)² - 4(2)(-6) = 16 + 48 = 64, which indicates the presence of two distinct real roots.
Solve by completing the square: x² + 6x + 5 = 0. What is one of the solutions?
-1
1
5
-3
Rewriting x² + 6x + 5 = 0 by completing the square gives (x + 3)² = 4, leading to solutions x = -3 ± 2. This results in x = -1 or x = -5, and -1 is one of the valid solutions.
Which of the following represents the vertex form of a quadratic function?
y = a(x + h)² - k
y = ax² + bx + c
y = a(x - h)² + k
y = (x - h)² + k
The vertex form, y = a(x - h)² + k, clearly identifies the vertex (h, k) of the parabola and the coefficient a which affects the graph's stretch. The other forms do not explicitly reveal the vertex's location.
For the quadratic function f(x) = 3(x-2)² + 4, what is the axis of symmetry?
x = 4
x = 2
x = -2
y = 4
The function is given in vertex form with the vertex at (2, 4), meaning the vertical line x = 2 is the axis of symmetry. This line divides the parabola into two mirror-image halves.
A quadratic equation has a discriminant of 0. What can be said about its roots?
It has two complex roots.
It has no real roots.
It has two distinct real roots.
It has one real repeated root.
A discriminant of 0 indicates that the quadratic equation touches the x-axis at a single point. This means the equation has one real repeated root, sometimes referred to as a double root.
When graphing a quadratic function, what does the coefficient 'a' determine?
The y-intercept
The x-intercepts of the parabola
The location of the vertex
The direction and width (stretch) of the parabola
The coefficient 'a' in a quadratic function influences whether the parabola opens upward or downward and also affects its width. A larger absolute value of a makes the parabola narrower, while a smaller value makes it wider.
How does the graph of a quadratic function change when the constant term 'c' is altered?
It shifts the graph horizontally.
It changes the width of the parabola.
It affects the direction the parabola opens.
It shifts the graph vertically.
The constant term 'c' in the quadratic function represents the y-intercept. Changing c results in a vertical shift of the graph up or down, without affecting its horizontal position or shape.
Solve the quadratic equation by factoring: x² - 5x + 6 = 0. What is the sum of its solutions?
3
6
2
5
Factoring x² - 5x + 6 gives (x - 2)(x - 3) = 0, leading to solutions x = 2 and x = 3. The sum of these solutions, 2 + 3, is 5, which also aligns with Vieta's formula.
Which quadratic equation represents a parabola that opens downward?
y = 2x² + 3x + 1
y = x² - 3x + 2
y = -2x² + 4x - 1
y = 2x² - 4x + 1
A parabola opens downward when the coefficient of x² is negative. In the equation y = -2x² + 4x - 1, the coefficient -2 confirms that the graph opens downward.
What transformation does the graph of y = (x-3)² + 2 represent compared to y = x²?
It shifts 3 units to the right and 2 units downward.
It shifts 3 units to the right and 2 units upward.
It shifts 3 units to the left and 2 units downward.
It shifts 3 units to the left and 2 units upward.
In the vertex form, y = (x-3)² + 2 indicates a horizontal shift 3 units to the right and a vertical shift 2 units upward relative to the basic parabola y = x². This transformation moves the vertex from (0, 0) to (3, 2).
Given the quadratic function f(x) = 2x² - 8x + k, determine the value of k such that the parabola touches the x-axis.
8
4
0
16
For the parabola to touch the x-axis, its discriminant must be zero. Setting b² - 4ac = (-8)² - 4(2)(k) = 64 - 8k equal to 0 results in k = 8, ensuring a single repeated real root.
Which method is most efficient for finding the vertex of a quadratic function given in standard form, and why?
Graphing by plotting many points.
Factoring the quadratic because it always reveals the vertex.
Using the vertex formula x = -b/(2a) because it directly computes the vertex's x-coordinate.
Using the quadratic formula, as it lists the vertex coordinates.
The vertex formula x = -b/(2a) is a direct and efficient method for finding the x-coordinate of the vertex from the standard form of a quadratic function. It avoids the more time-consuming processes of factoring or plotting numerous points.
Find the axis of symmetry for the quadratic equation 5x² + 20x + 15 = 0 using the formula. What is the correct equation of the axis?
x = -4
x = 4
x = 2
x = -2
The axis of symmetry for a quadratic equation in standard form is given by x = -b/(2a). Substituting a = 5 and b = 20 yields x = -20/(10) = -2.
If a quadratic function in vertex form is given by y = -3(x + 1)² + 7, what is the maximum value of the function?
1
-7
7
The function has no maximum.
The vertex form of the quadratic function reveals the vertex, which in this case is (-1, 7). Because the coefficient -3 is negative, the parabola opens downward and the vertex represents the maximum value, which is 7.
Determine the solutions of the quadratic equation 3x² - 2x - 5 = 0 using the quadratic formula.
x = 1 or x = 5
x = 2 or x = -5/2
x = 5/3 or x = -1
x = -5/3 or x = 1
Applying the quadratic formula with a = 3, b = -2, and c = -5 produces a discriminant of 64. This yields the solutions x = (2 ± 8) / 6, simplifying to x = 5/3 and x = -1.
0
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Study Outcomes

  1. Analyze quadratic equations using factoring, completing the square, and the quadratic formula.
  2. Graph quadratic functions by identifying the vertex, axis of symmetry, and intercepts.
  3. Evaluate how changes in coefficients affect the shape and position of quadratic graphs.
  4. Apply algebraic techniques to solve quadratic equations in various problem contexts.

Quadratic Functions Quiz Answers Cheat Sheet

  1. Quadratic Formula - This go‑to tool cracks any quadratic ax² + bx + c = 0 in a snap, thanks to x = ( - b ± √(b² - 4ac)) / (2a). Practice plugging in values until it feels like second nature! Dive deeper
  2. Discriminant - The expression b² - 4ac is your root‑detective: positive means two real roots, zero gives one real root, and negative reveals complex roots. Understanding this saves you time and surprises on tests! Study the details
  3. Graphing Parabolas - Parabolas are U‑shaped curves with key features: the vertex (peak or valley), the axis of symmetry, and the direction it opens. Sketching them helps visualize how changes in a, h, and k transform the curve! See examples
  4. Factoring Quadratics - Breaking x² + bx + c into (x - r₝)(x - r₂) makes solving a breeze. Spotting factors quickly grows easier with pattern practice and boosts your confidence! Get factoring tips
  5. Completing the Square - Turn ax² + bx + c into a perfect square trinomial to solve when factoring fails. It's a powerful tool that also leads you right to the vertex form! Learn the steps
  6. Transformations - Shifting, stretching, and reflecting f(x)=a(x - h)²+k changes your parabola's position and shape. Play with h and k to see horizontal and vertical moves in action! Explore transformations
  7. Vertex Form - f(x)=a(x - h)²+k makes spotting the vertex (h,k) a breeze and shows exactly how your graph shifts. It's the quickest way to graph with precision! Master vertex form
  8. Axis of Symmetry - This vertical line x = - b/(2a) slices your parabola into mirror halves and always passes through the vertex. It's a hidden guideline for plotting points symmetrically! Find out more
  9. Zero Product Property - If (factor₝)(factor₂)=0, then factor₝ or factor₂ must be zero - boom, instant solutions! It's the backbone of solving factored quadratics. Review this rule
  10. Real‑World Applications - Quadratics pop up in projectile motion, profit calculations, and more. Applying them to real scenarios makes the math stick and feels downright practical! See examples
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