Quadratic Vertex Form Formula

Quadratic vertex form is a quadratic written as f(x) = a(x - h)^2 + k, where the vertex (h, k) is directly readable from the formula.

The Formula

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

When to use: Imagine sliding a basic x2x^2 parabola around on the coordinate plane. The value hh shifts it left or right, kk shifts it up or down, and aa stretches or flips it. The vertex (h,k)(h, k) is the parabola's turning pointβ€”you can read it directly from this form.

Quick Example

f(x)=2(xβˆ’3)2+1f(x) = 2(x - 3)^2 + 1 The vertex is (3,1)(3, 1), the parabola opens upward, and it is narrower than x2x^2.

Notation

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

What This Formula Means

A quadratic written as f(x)=a(xβˆ’h)2+kf(x) = a(x - h)^2 + k, where the vertex (h,k)(h, k) is directly readable from the formula.

Imagine sliding a basic x2x^2 parabola around on the coordinate plane. The value hh shifts it left or right, kk shifts it up or down, and aa stretches or flips it. The vertex (h,k)(h, k) is the parabola's turning pointβ€”you can read it directly from this form.

Formal View

f(x)=a(xβˆ’h)2+kf(x) = a(x - h)^2 + k with aβ‰ 0a \neq 0, where (h,k)=(βˆ’b2a,β€…β€Šcβˆ’b24a)(h, k) = \left(-\frac{b}{2a},\; c - \frac{b^2}{4a}\right). The vertex is the global extremum: minimum if a>0a > 0, maximum if a<0a < 0, with f(h)=kf(h) = k.

Worked Examples

Example 1

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

Answer

(3,1)(3, 1)

First step

1
Vertex form is a(xβˆ’h)2+ka(x - h)^2 + k with vertex (h,k)(h, k).

Full solution

  1. 2
    Here h=3h = 3 and k=1k = 1.
  2. 3
    The vertex is (3,1)(3, 1).
In vertex form, the vertex coordinates are read directly: hh is the value subtracted from xx, and kk is the constant added.

Example 2

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

Example 3

easy
Write the vertex form of a parabola with vertex (2,βˆ’3)(2, -3) and a=1a = 1.

Common Mistakes

  • Taking hh with the wrong sign - (xβˆ’h)(x-h) means hh is the value that makes the inside zero, so (x+3)2(x+3)^2 has h=βˆ’3h=-3.
  • Forgetting aa also flips/stretches - a negative aa opens the parabola downward (vertex is a maximum).
  • Reading (h,k)(h,k) from standard form - convert to a(xβˆ’h)2+ka(x-h)^2+k first; the vertex is not the bb and cc.

Why This Formula Matters

It hands you the parabola's max or min and its axis of symmetry for free, which is exactly what optimization and graphing questions want. Recognizing the minus sign in (xβˆ’h)(x-h) is the difference between a right and a backwards graph. Recognizing it by "Is the quadratic written as a squared binomial plus a constant, and do I want its turning point?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from standard form and factored form and completing the square in a mixed problem set.

Frequently Asked Questions

What is the Quadratic Vertex Form formula?

A quadratic written as f(x)=a(xβˆ’h)2+kf(x) = a(x - h)^2 + k, where the vertex (h,k)(h, k) is directly readable from the formula.

How do you use the Quadratic Vertex Form formula?

Imagine sliding a basic x2x^2 parabola around on the coordinate plane. The value hh shifts it left or right, kk shifts it up or down, and aa stretches or flips it. The vertex (h,k)(h, k) is the parabola's turning pointβ€”you can read it directly from this form.

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

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

Why is the Quadratic Vertex Form formula important in Math?

It hands you the parabola's max or min and its axis of symmetry for free, which is exactly what optimization and graphing questions want. Recognizing the minus sign in (xβˆ’h)(x-h) is the difference between a right and a backwards graph. Recognizing it by "Is the quadratic written as a squared binomial plus a constant, and do I want its turning point?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from standard form and factored form and completing the square in a mixed problem set.

What do students get wrong about Quadratic Vertex Form?

The procedure for quadratic vertex form is the easy part; the trap is taking hh with the wrong sign. Asking "Is the quadratic written as a squared binomial plus a constant, and do I want its turning point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 β†’
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