Quadratic Vertex Form Formula

The Formula

f(x) = a(x - h)^2 + k with vertex at (h, k)

When to use: 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.

Quick Example

f(x) = 2(x - 3)^2 + 1 The vertex is (3, 1), the parabola opens upward, and it is narrower than x^2.

Notation

a(x - h)^2 + k where h is the horizontal shift (note the minus sign!) and k is the vertical shift. When a > 0 the parabola opens upward; when a < 0 it opens downward.

What This Formula Means

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.

Formal View

f(x) = a(x - h)^2 + k with a \neq 0, where (h, k) = \left(-\frac{b}{2a},\; c - \frac{b^2}{4a}\right). The vertex is the global extremum: minimum if a > 0, maximum if a < 0, with f(h) = k.

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).

Common Mistakes

  • Getting the sign of h wrong—in (x + 2)^2 the vertex is at x = -2, not x = 2
  • Forgetting to multiply a back in when converting from standard form
  • Confusing vertex form with factored form

Why This Formula Matters

Vertex form lets you read the maximum or minimum value of a quadratic instantly, which is critical for optimization problems in physics and business. It also makes graphing effortless since the vertex and direction of opening are immediately visible.

Frequently Asked Questions

What is the Quadratic Vertex Form formula?

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.

How do you use the Quadratic Vertex Form formula?

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.

What do the symbols mean in the Quadratic Vertex Form formula?

a(x - h)^2 + k where h is the horizontal shift (note the minus sign!) and k is the vertical shift. When a > 0 the parabola opens upward; when a < 0 it opens downward.

Why is the Quadratic Vertex Form formula important in Math?

Vertex form lets you read the maximum or minimum value of a quadratic instantly, which is critical for optimization problems in physics and business. It also makes graphing effortless since the vertex and direction of opening are immediately visible.

What do students get wrong about Quadratic Vertex Form?

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

What should I learn before the Quadratic Vertex Form formula?

Before studying the Quadratic Vertex Form formula, you should understand: quadratic functions, quadratic standard form.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
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