# 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

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