Question

How do you rewrite into vertex form `f(x)=4x^2-16x+5`?

Answer 1

Please see the explanation.

Explanation:

The vertex form for a parabola that opens or down is:

f(x) = a(x - h)^2 + k

Please observe that, in the given equation, `a = 4`, therefore we add zero to the original equation in the form of `4h^2 - 4h^2`:

`f(x) = 4x^2 - 16x + 4h^2 - 4h^2 + 5`

Factor 4 out of the first 3 terms and group them with ()s:

`f(x) = 4(x^2 - 4x + h^2) - 4h^2 + 5`

We can find the value of h by setting the middle term equal to -2hx:

`-2hx = -4x`

`h = 2` and `-4h^2 = -16`

`f(x) = 4(x^2 - 4x + h^2) - 16 + 5`

We know that three terms inside the ()s is equal to `(x - 2)^2`

`f(x) = 4(x - 2)^2 - 11`