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Quadratic Functions and Graphs

A free College Algebra lesson from the “Quadratic Functions and Equations” unit, with a worked example and practice problems including step-by-step solutions.

Quadratic graphs are parabolas. Vertex form reveals the vertex, while factored form reveals zeros. The leading coefficient controls opening direction.

What you'll learn

Why it matters: Projectile paths, revenue curves, bridge arches, and reflector shapes all use parabolas. The vertex, axis, zeros, and opening direction tell students what the graph is doing before they calculate every point.

Worked example

Problem. Find the vertex of y = (x - 2)^2 - 9.

  1. Use y = a(x - h)^2 + k.
  2. h = 2 and k = -9.
  3. The vertex is (2, -9).

Answer: (2, -9)

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. The vertex of y = (x + 4)^2 + 1 is...

Choices: (-4, 1) · (4, 1) · (-4, -1) · (4, -1)

Show solution
  1. x + 4 means h = -4.

Answer: (-4, 1)

2. Does y = -2x^2 + 5 open up or down?

Choices: Down · Up · Left · Right

Show solution
  1. The leading coefficient is negative.

Answer: Down

3. A quadratic has zeros 2 and 10. What is its axis of symmetry?

Show solution
  1. Average the zeros.

Answer: 6

4. The vertex of y = (x + 5)^2 + 1 is...

Choices: (-5, 1) · (5, 1) · (-5, -1) · (1, -5)

Show solution
  1. Vertex form y = (x - h)^2 + k gives (h, k).

Answer: (-5, 1)

5. For y = (x - 4)^2 - 7, what is the axis of symmetry? Enter the x-value.

Show solution
  1. The axis passes through the vertex.

Answer: 4

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