Answer
The vertex is $(-3,-1)$.
The vertical intercept is $(0,8)$.
The horizontal intercepts are $(-2,0)$ and $(-4,0)$.
Work Step by Step
The given function is
$\Rightarrow f(x)=x^2+6x+8$
The quadratic function in standard form is
$\Rightarrow f(x)=ax^2+bx+c$
$a=1,b=6$ and $c=8$
Vertex:-
Input values is
$x=\frac{-b}{2a}$
Substitute all values.
$x=\frac{-6}{2(1)}$
Simplify.
$x=-3$
For the output value substitute $x=-3$ into given function.
$\Rightarrow f(-3)=(-3)^2+6(-3)+8$
Clear the parentheses.
$\Rightarrow f(-3)=9-18+8$
Simplify.
$\Rightarrow f(-3)=-1$
The vertex is $(-3,-1)$.
Vertical intercept :-
Substitute $x=0$ into the given function.
$\Rightarrow f(0)=(0)^2+6(0)+8$
Clear the parentheses.
$\Rightarrow f(0)=8$
The vertical intercept is $(0,8)$.
Horizontal intercept(s):-
Substitute $f(x)=0$ into the given function.
$\Rightarrow 0=x^2+6x+8$
Use quadratic formula,
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute $a=1,b=6$ and $c=8$ into the formula.
$x=\frac{-(6)\pm\sqrt{(6)^2-4(1)(8)}}{2(1)}$
Simplify.
$x=\frac{-6\pm\sqrt{36-32}}{2}$
$x=\frac{-6\pm\sqrt{4}}{2}$
$x=\frac{-6\pm2}{2}$
Separate the equations.
$x=\frac{-6+2}{2}$ and $x=\frac{-6-2}{2}$
Simplify.
$x=\frac{-4}{2}$ and $x=\frac{-8}{2}$
$x=-2$ and $x=-4$
The horizontal intercepts are $(-2,0)$ and $(-4,0)$.
