Convert a quadratic function from the vertex form to the general form

We show you in this lesson how to convert a quadratic function from the vertex form to the general form. 

The vertex form is f(x) = a(x - h)2 + k and the general form is f(x) = ax2 + bx + c

Using f(x) = a(x - h)2 + k, you can just expand the function

f(x) = a(x - h)2 + k

f(x) = a(x - h)(x - h) + k

f(x) = a(x2 - hx - hx + h2) + k

f(x) = a(x2 - 2hx + h2) + k

f(x) = ax2 - 2ahx + ah2 + k

Method #1

You can then use f(x) = ax2 - 2ahx + ah2 + k as a sort of "formula"

Comparing the expression above with f(x) = ax2 + bx + c, we notice the following:

a = a

b = - 2ah

c = ah2 + k

Suppose y = 2(x + 3)2 + 1

Rewrite in vertex form: y = 2(x - -3)2 + 1

a = 2

b = - 2ah = -2(2)(-3) = (-4)(-3) =12

c = ah2 + k = 2(-3)2 + 1 = 2(9) + 1 = 19

f(x) = ax2 + bx + c = 2x2 + 12x + 19

Method #2

Instead of using the "formula" or f(x) = ax2 - 2ahx + ah2 + k, you can just expand y = 2(x + 3)2 + 1.

y = 2(x + 3)2 + 1

y = 2(x + 3)(x + 3) + 1

y = 2(x2 + 3x + 3x + 9) + 1

y = 2(x2 + 6x + 9) + 1

y = 2x2 + 12x + 18 + 1

y = 2x2 + 12x + 19

Recent Articles

  1. Write a Polynomial from Standard Form to Factored Form

    Oct 14, 21 05:41 AM

    Learn how to write a polynomial from standard form to factored form

    Read More

Enjoy this page? Please pay it forward. Here's how...

Would you prefer to share this page with others by linking to it?

  1. Click on the HTML link code below.
  2. Copy and paste it, adding a note of your own, into your blog, a Web page, forums, a blog comment, your Facebook account, or anywhere that someone would find this page valuable.