# F(x) = x^2+6x+8?

Oct 19, 2017

Note that this is not a question; it is just a definition.

#### Explanation:

Some possible questions:

1. What are the $x$ and $y$ intercepts for this equation?
a) The $y$ intercept is the value of the equation when $x = 0$
In this case if $x = 0$
$\textcolor{w h i t e}{\text{XXX}} F \left(0\right) = {0}^{2} + 6 \times 0 + 8 = 8$
So the $y$ intercept is $8$

b) The $x$ intercepts are the values of $x$ for which the equation is equal to $0$
In this case if ${x}^{2} + 6 x + 8 = 0$
$\textcolor{w h i t e}{\text{XXX}}$We can factor the expression on the left:
$\textcolor{w h i t e}{\text{XXX}} \left(x + 2\right) \left(x + 4\right) = 0$
color(white)("XXX")rarr{:((x+2)=0,color(white)("xx")"or"color(white)("xx"),(x+4)=0), (rarr x=-2,,rarr x=-4) :}
So the $x$ intercepts are $\left(- 2\right)$ and $\left(- 4\right)$

2. What is the vertex of this equation?
We can convert this equation into vertex form: $F \left(x\right) = {\left(x - \textcolor{red}{a}\right)}^{2} + \textcolor{b l u e}{b}$ with vertex at $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$
$\textcolor{w h i t e}{\text{XXX}} F \left(x\right)$
$\textcolor{w h i t e}{\text{XXXXX}} = {x}^{2} + 6 x + 8$

$\textcolor{w h i t e}{\text{XXXXX}} = {x}^{2} + 6 x + 9 - 1$

$\textcolor{w h i t e}{\text{XXXXX}} = {\left(x + 3\right)}^{2} - 1$

$\textcolor{w h i t e}{\text{XXXXX}} = {\left(x - \textcolor{red}{\left(- 3\right)}\right)}^{2} + \textcolor{b l u e}{\left(- 1\right)}$
which is the vertex form with vertex at $\left(\textcolor{red}{\left(- 3\right)} , \textcolor{b l u e}{\left(- 1\right)}\right)$

3. Draw the graph of this equation
graph{x^2+6x+8 [-8.836, 2.266, -1.523, 4.024]}