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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