91Ó°ÊÓ

Consider the quadratic function $f\left(x\right)=-4x^2-6x-2$.

For this parabola, the graph will open downwards as the leading coefficient is negative.

Then finding the axis of symmetry, vertex, the intercepts, and a point with help of the axis of symmetry, will help us to plot the points of the graph and get the required graph.

On comparing the given function $f\left(x\right)=-4x^2-6x-2$with the standard form $f\left(x\right)=a{x}^2+bx+c$we get

$$\begin{gathered} a=-4 \\ b=-6 \\ c=-2 \end{gathered}$$

The axis of symmetry for the standard form is $x=-\frac{b}{2a}$.

So for the given function, the axis of symmetry is given as

role="math" localid="1645161353854" $$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{-6}{2·(-4)} \\ x=-\frac{3}{4} \\ x=-0.75 \end{gathered}$$

Thus the equation of the axis of symmetry is $x=-0.75$.

Vertex is a point on the line of axis of symmetry.

Here axis of symmetry is $x=-0.75$.

So the function value $f(-0.75)$is given as

$$\begin{gathered} f(-0.75)=-4·{(-0.75)}^2-6·(-0.75)-2 \\ f(-0.75)=-2.25+4.5-2 \\ f(-0.75)=0.25 \end{gathered}$$

So the vertex of the parabola is $(-0.75,0.25)$.

To find the y-intercept substitute $0$for $x$in the function.

$$\begin{gathered} f\left(0\right)=-4·0^2-6·0-2 \\ f\left(0\right)=-2 \end{gathered}$$

Thus the y intercept is $(0,-2)$.

To find $x$intercepts substitute $0$for $f\left(x\right)$in the function.

role="math" localid="1645161527000" $\begin{array}{rcl}f\left(x\right)& =& -4x^2-6x-2\\ 0& =& -4x^2-6x-2\end{array}$

Factoring the trinomial we get

$$\begin{gathered} 0=-4x^2-4x-2x-2 \\ 0=-4x(x+1)-2(x+1) \\ 0=(x+1)(-4x-2) \end{gathered}$$

So the solutions are

$\begin{array}{rcl}x+1& =& 0\\ x& =& -1\end{array}$

and

$\begin{array}{rcl}-4x-2& =& 0\\ -4x& =& 2\\ x& =& -0.5\end{array}$

So thex intercepts are $(-1,0),(-0.5,0)$.

We have y intercept as $(0,-2)$.

The axis of symmetry is$x=-0.75$.
It means that the intercept is at a distance of$0.75$ units to the right of the axis of symmetry.
We can get a point on the graph using this information.
A point at the same distance, i.e., $0.75$units, but to the left will also lie on the graph. This point will be $(-1.5,-2)$.

Plotting all the points we got and joining these points will give us the graph as-