91Ó°ÊÓ

We have:

$f\left(x\right)=-2x^2-8x-12$

First, calculate the $x-$intercept by putting $f\left(x\right)=0$.

$$\begin{gathered} f\left(x\right)=-2x^2-8x-12 \\ -2x^2-8x-12=0 \end{gathered}$$

$$\begin{gathered} \mathrm{For} \mathrm{a} \mathrm{quadratic} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{form} a{x}^2+bx+c=0 \mathrm{the} \mathrm{solutions} \mathrm{are}  \\ {x}_{1, 2}=\frac{-bÂ\pm \sqrt{b^2-4ac}}{2a} \\ \mathrm{For} a=-2, b=-8, c=-12 \\ {x}_{1, 2}=\frac{-\left(-8\right)Â\pm \sqrt{{\left(-8\right)}^2-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)} \\ {x}_{1, 2}=\frac{-\left(-8\right)Â\pm  4\sqrt{2}i}{2\left(-2\right)} \\ x=-2-\sqrt{2}i, x=-2+\sqrt{2}i \end{gathered}$$

There is no $x-$intercept: $x∉R$

Now, calculate $y-$intercept by putting role="math" localid="1645247299931" $x=0$:

$$\begin{gathered} y=-2{\left(0\right)}^2-8\left(0\right)-12 \\ y=-12 \end{gathered}$$

$y-$intercept:$\left(0,-12\right)$

$$\begin{gathered} \mathrm{The} \mathrm{vertex} \mathrm{of} \mathrm{an} \mathrm{up}-\mathrm{down} \mathrm{facing} \mathrm{parabola} \mathrm{of} \mathrm{the} \mathrm{form} y=a{x}^2+bx+c \mathrm{is} x_v=-\frac{b}{2a} \\ \mathrm{The} \mathrm{parabola} \mathrm{params} \mathrm{are}: \\ a=-2, b=-8, c=-12 \\ {x}_v=-\frac{b}{2a} \\ {x}_v=-\frac{\left(-8\right)}{2\left(-2\right)} \\ {x}_v=-2 \end{gathered}$$

Plug $x_v=-2$to find $y_v$:

$$\begin{gathered} y_v=-4 \\ \mathrm{Therefore} \mathrm{the} \mathrm{parabola} \mathrm{vertex} \mathrm{is} \\ \left(-2, -4\right) \\ \mathrm{If} \mathrm{a}<0, \mathrm{then} \mathrm{the} \mathrm{vertex} \mathrm{is} \mathrm{a} \mathrm{maximum} \mathrm{value} \\ \mathrm{If} \mathrm{a}>0, \mathrm{then} \mathrm{the} \mathrm{vertex} \mathrm{is} \mathrm{a} \mathrm{minimum} \mathrm{value} \\ \mathrm{a}=-2 \\ \mathrm{Maximum}\left(-2, -4\right) \end{gathered}$$

We have:

$f\left(x\right)=-2x^2-8x-12$

Identify the coefficients:

$a=-1,b=-8$

Substitute the coefficients into the expression:

$$\begin{gathered} x=-\frac{-8}{2×(-2)} \\ x=-2 \end{gathered}$$

Plot the y-intercepts on the graph then plot the vertex at $(-2,-4)$and then draw the axis of symmetry: