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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