Quadratic Vertex Form Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadratic Vertex Form.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and a determines its width and direction.

Imagine sliding a basic x^2 parabola around on the coordinate plane. The value h shifts it left or right, k shifts it up or down, and a stretches or flips it. The vertex (h, k) is the parabola's turning point—you can read it directly from this form.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Vertex form reveals the most important graphing information at a glance: the vertex location and the direction of opening.

Common stuck point: The sign convention: a(x - h)^2 + k has a minus sign, so f(x) = (x + 2)^2 means h = -2, not h = 2.

Sense of Study hint: When you see a(x - h)^2 + k, read the vertex directly as (h, k) — but watch the minus sign inside the parentheses. First identify a to determine the opening direction, then plot the vertex and a few symmetric points on each side. Finally, sketch the parabola through those points.

Worked Examples

Example 1

easy
What is the vertex of f(x) = 2(x - 3)^2 + 1?

Solution

  1. 1
    Vertex form is a(x - h)^2 + k with vertex (h, k).
  2. 2
    Here h = 3 and k = 1.
  3. 3
    The vertex is (3, 1).

Answer

(3, 1)
In vertex form, the vertex coordinates are read directly: h is the value subtracted from x, and k is the constant added.

Example 2

medium
Write the vertex form of a parabola with vertex (-1, 4) passing through (0, 7).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What is the vertex of g(x) = -(x + 5)^2 - 2?

Example 2

hard
Convert x^2 + 4x + 7 to vertex form.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionsquadratic standard form