# How do you write f(x)= -2x^2 + 16x +4 in vertex form?

Mar 26, 2016

$f \left(x\right) = - 2 {\left(x - 4\right)}^{2} + 36$ with vertex at $\left(4 , 36\right)$

#### Explanation:

General vertex form is
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with the vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$

Given
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = - 2 {x}^{2} + 16 x + 4$

Extract the $\textcolor{g r e e n}{m}$ component:
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = \textcolor{g r e e n}{- 2} \left({x}^{2} - 8 x\right) + 4$

Complete the square:
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = \textcolor{g r e e n}{- 2} \left({x}^{2} - 8 x + \textcolor{c y a n}{16}\right) + 4 - \textcolor{c y a n}{\left(- 2 \cdot\right) \left(16\right)}$

Re-write as squared binomial and simplify to get vertex form
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = \textcolor{g r e e n}{- 2} {\left(x - \textcolor{red}{4}\right)}^{2} + \textcolor{b l u e}{36}$
graph{-2x^2+16x+4 [-5.92, 26.13, 22.5, 38.54]}

The vertex form is ${\left(x - 4\right)}^{2} = - \frac{1}{2} \cdot \left(y - 36\right)$

#### Explanation:

We start from the given $f \left(x\right) = - 2 {x}^{2} + 16 x + 4$

Let $y = - 2 {x}^{2} + 16 x + 4$

Start by factoring out the -2 from the first two terms

$y = - 2 \left({x}^{2} - 8 x\right) + 4$

We now use the -8. Divide this number by 2 then the result be squared so that we will have ${\left(- \frac{8}{2}\right)}^{2} = + 16$

This 16 will be added and subtracted inside the grouping symbol.

$y = - 2 \left({x}^{2} - 8 x\right) + 4$

$y = - 2 \left({x}^{2} - 8 x + 16 - 16\right) + 4$

We now have a PFT-Perfect Square Trinomial $\left({x}^{2} - 8 x + 16\right) = {\left(x - 4\right)}^{2}$

So that we have

$y = - 2 \left({x}^{2} - 8 x + 16 - 16\right) + 4$

$y = - 2 \left({\left(x - 4\right)}^{2} - 16\right) + 4$

Put the -2 back

$y = - 2 {\left(x - 4\right)}^{2} + 32 + 4$

Simplify

$y = - 2 {\left(x - 4\right)}^{2} + 36$
transpose the 36 to the left of the equation

$y - 36 = - 2 {\left(x - 4\right)}^{2}$

divide by -2

${\left(x - 4\right)}^{2} = - \frac{1}{2} \cdot \left(y - 36\right)$

God bless....I hope the explanation is useful.

Mar 26, 2016

$y = - 2 {\left(x - 4\right)}^{2} + 36$

#### Explanation:

There is another way of finding the vertex form.
x-coordinate of vertex:
$x = - \frac{b}{2 a} = - \frac{16}{-} 4 = 4$
y-coordinate of vertex:
y(4) = -32 + 64 + 4 = 36
Vertex form:
$y = a {\left(x - 4\right)}^{2} + 36 = - 2 {\left(x - 4\right)}^{2} + 36$