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Vertex Form of Quadratics

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

Vertex form writes a quadratic as y = a(x - h)^2 + k. This form shows the vertex directly: (h, k). The vertex is the turning point of the parabola, so it gives the maximum or minimum value depending on whether the graph opens down or up. Vertex form matters because it makes transformations visible: h shifts the graph left or right, k shifts it up or down, and a changes width and direction. A common mistake is reading h with the wrong sign; in (x - h), the x-coordinate is h.

What you'll learn

Why it matters: Projectile maximum heights, parabolic reflectors, satellite dishes, and revenue-curve optima are read directly from vertex form. The vertex is the answer to 'what is the best or worst value, and where does it happen?'

Worked example

Problem. Find the vertex of y = -2(x - 4)^2 + 7.

  1. Compare with y = a(x - h)^2 + k.
  2. h = 4 and k = 7.
  3. The vertex is (4, 7).

Answer: (4, 7)

Practice problems

1. The vertex of y = (x - 3)^2 + 5 is...

Choices: (3, 5) · (-3, 5) · (3, -5) · (-3, -5)

Show solution
  1. Warm-up: First identify exactly what the question is asking: The vertex of y = (x - 3)^2 + 5 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Vertex form 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: (3, 5)

2. The vertex of y = (x + 6)^2 - 2 is...

Choices: (-6, -2) · (6, -2) · (-6, 2) · (6, 2)

Show solution
  1. Warm-up: First identify exactly what the question is asking: The vertex of y = (x + 6)^2 - 2 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. x + 6 means h = -6.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-6, -2)

3. Does y = -3(x - 1)^2 + 8 open up or down?

Choices: Down · Up · Left · Right

Show solution
  1. Core Practice: First identify exactly what the question is asking: Does y = -3(x - 1)^2 + 8 open up or down?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. a 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

4. For y = (x - 9)^2 - 4, what is the axis of symmetry?

Show solution
  1. Core Practice: First identify exactly what the question is asking: For y = (x - 9)^2 - 4, what is the axis of symmetry?
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. The axis is x = h.
  4. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

5. For y = -2(x + 5)^2 + 11, the maximum value is...

Choices: 11 · -5 · 2 · -2

Show solution
  1. Challenge: First identify exactly what the question is asking: For y = -2(x + 5)^2 + 11, the maximum value is...
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Opening down means the vertex y-value is the maximum.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 11

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