Answer
The vertex is $(-6,-64)$.
The vertical intercept is $(0,-28)$.
The horizontal intercepts are $(2,0)$ and $(-14,0)$.
Work Step by Step
The given function is
$\Rightarrow g(x)=x^2+12x-28$
The quadratic function in standard form is
$\Rightarrow f(x)=ax^2+bx+c$
$a=1,b=12$ and $c=-28$
Vertex:-
Input values is
$x=\frac{-b}{2a}$
Substitute all values.
$x=\frac{-12}{2(1)}$
Simplify.
$x=-6$
For the output value substitute $x=-3$ into given function.
$\Rightarrow f(-6)=(-6)^2+12(-6)-28$
Clear the parentheses.
$\Rightarrow f(-6)=36-72-28$
Simplify.
$\Rightarrow f(-6)=-64$
The vertex is $(-6,-64)$.
Vertical intercept :-
Substitute $x=0$ into the given function.
$\Rightarrow f(0)=(0)^2+12(0)-28$
Clear the parentheses.
$\Rightarrow f(0)=-28$
The vertical intercept is $(0,-28)$.
Horizontal intercept(s):-
Substitute $f(x)=0$ into the given function.
$\Rightarrow 0=x^2+12x-28$
Use quadratic formula,
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute $a=1,b=12$ and $c=-28$ into the formula.
$x=\frac{-(12)\pm\sqrt{(12)^2-4(1)(-28)}}{2(1)}$
Simplify.
$x=\frac{-12\pm\sqrt{144+112}}{2}$
$x=\frac{-12\pm\sqrt{256}}{2}$
$x=\frac{-12\pm16}{2}$
Separate the equations.
$x=\frac{-12+16}{2}$ and $x=\frac{-12-16}{2}$
Simplify.
$x=\frac{4}{2}$ and $x=\frac{-28}{2}$
$x=2$ and $x=-14$
The horizontal intercepts are $(2,0)$ and $(-14,0)$.
