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Quiz: Quadratic Functions and Equations

Test your understanding of quadratic functions, parabolas, and methods for solving quadratic equations with these questions.

1. What is the graph of a quadratic function called?

  1. A line
  2. A parabola
  3. A hyperbola
  4. A circle
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The correct answer is B. The graph of any quadratic function f(x) = ax² + bx + c is a U-shaped curve called a parabola. It opens upward if a > 0 or downward if a < 0. Option A describes linear functions. Options C and D are different conic sections.

Concept Tested: Parabola

See: Parabolas and Their Key Features

2. In the quadratic function f(x) = x² - 6x + 5, what is the axis of symmetry?

  1. x = -6
  2. x = 6
  3. x = 3
  4. x = -3
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The correct answer is C. The axis of symmetry is found using x = -b/(2a). Here, a = 1 and b = -6, so x = -(-6)/(2·1) = 6/2 = 3. This vertical line passes through the vertex and divides the parabola into two mirror images. Option A uses b directly. Option B forgets to divide by 2. Option D uses the wrong sign.

Concept Tested: Axis of Symmetry

See: Axis of Symmetry

3. What does the discriminant b² - 4ac tell you about a quadratic equation?

  1. The y-intercept
  2. The vertex location
  3. The nature and number of roots
  4. The axis of symmetry
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The correct answer is C. The discriminant Δ = b² - 4ac determines the nature of roots: if Δ > 0, there are two distinct real roots; if Δ = 0, one real root (repeated); if Δ < 0, no real roots (two complex roots). Options A, B, and D describe other features found using different formulas.

Concept Tested: Discriminant / Nature of Roots

See: The Discriminant

4. Solve x² - 9 = 0 using the square root method.

  1. x = 9
  2. x = ±3
  3. x = 3 only
  4. x = ±9
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The correct answer is B. Rearrange to x² = 9, then take the square root of both sides: x = ±√9 = ±3. Both x = 3 and x = -3 are solutions. Option A forgets the ± symbol. Option C gives only the positive root. Option D doesn't take the square root.

Concept Tested: Solving by Square Roots

See: Solving by Using Square Roots

5. What is the vertex form of a quadratic function?

  1. f(x) = ax² + bx + c
  2. f(x) = a(x - h)² + k
  3. f(x) = a(x - r₁)(x - r₂)
  4. y = mx + b
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The correct answer is B. Vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes the vertex immediately visible. Option A is standard form. Option C is factored form (when roots exist). Option D is linear, not quadratic.

Concept Tested: Vertex Form

See: Vertex Form

6. Use the quadratic formula to solve x² + 5x + 3 = 0. What is the correct setup?

  1. x = (-5 ± √(25 - 12))/2
  2. x = (5 ± √(25 + 12))/2
  3. x = (-5 ± √(25 + 12))/2
  4. x = (-5 ± √(5 - 12))/2
Show Answer

The correct answer is A. The quadratic formula is x = (-b ± √(b² - 4ac))/(2a). Here, a = 1, b = 5, c = 3, so x = (-5 ± √(25 - 12))/2 = (-5 ± √13)/2. Option B uses +b instead of -b. Option C uses 4ac = +12 instead of -12 in the discriminant. Option D uses b instead of b².

Concept Tested: Quadratic Formula

See: The Quadratic Formula

7. A ball is thrown upward with height h(t) = -16t² + 48t + 5. When does it hit the ground?

  1. When h(t) = 0
  2. When t = 0
  3. When h(t) = 5
  4. When t = 48
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The correct answer is A. The ball hits the ground when its height is zero, so set h(t) = 0 and solve for t. This gives -16t² + 48t + 5 = 0. Using the quadratic formula yields t ≈ 3.1 seconds (the positive solution). Option B gives the initial time. Option C is the initial height. Option D misinterprets the coefficient.

Concept Tested: Projectile Motion / Applications of Quadratics

See: Projectile Motion

8. What does the imaginary unit i represent?

  1. √1
  2. √(-1)
  3. The x-intercept
  4. Infinity
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The correct answer is B. The imaginary unit i is defined as i = √(-1), which means i² = -1. Complex numbers use i to express square roots of negative numbers. For example, √(-4) = 2i. Option A equals 1, not i. Options C and D are unrelated concepts.

Concept Tested: Imaginary Unit / Complex Numbers

See: The Imaginary Unit

9. Solve by factoring: x² + 7x + 12 = 0.

  1. x = -3 or x = -4
  2. x = 3 or x = 4
  3. x = -3 or x = 4
  4. x = 3 or x = -4
Show Answer

The correct answer is A. Factor: (x + 3)(x + 4) = 0. By the zero product property, x + 3 = 0 or x + 4 = 0, giving x = -3 or x = -4. You can verify: (-3)² + 7(-3) + 12 = 9 - 21 + 12 = 0 ✓. Option B has the wrong signs. Options C and D mix correct and incorrect values.

Concept Tested: Solving by Factoring / Zero Product Property

See: Solving by Factoring

10. If a quadratic function has vertex (2, -5) and opens upward, what is the minimum value?

  1. 2
  2. -5
  3. -2
  4. 5
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The correct answer is B. For a parabola that opens upward, the vertex represents the minimum point. The minimum value is the y-coordinate of the vertex, which is -5. Option A gives the x-coordinate. Options C and D are incorrect transformations of the values.

Concept Tested: Minimum Value / Vertex

See: Maximum and Minimum Values