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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)

Practice problems

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

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: The vertex of y = (x + 4)^2 + 1 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. x + 4 means h = -4.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-4, 1)

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

Choices: Down · Up · Left · Right

Show solution
  1. Core Practice: First identify exactly what the question is asking: Does y = -2x^2 + 5 open up or down?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. The leading coefficient is negative.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Down

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

Show solution
  1. Challenge: First identify exactly what the question is asking: A quadratic has zeros 2 and 10. What is its axis of symmetry?
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Average the zeros.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

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: First identify exactly what the question is asking: The vertex of y = (x + 5)^2 + 1 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Vertex form y = (x - h)^2 + k gives (h, k).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-5, 1)

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

Show solution
  1. Axis of Symmetry: First identify exactly what the question is asking: For y = (x - 4)^2 - 7, what is the axis of symmetry? Enter the x-value.
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. The axis passes through the vertex.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

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