91Ó°ÊÓ

Consider the quadratic function $f\left(x\right)=-16x^2+24x-9$.

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 $f\left(x\right)=-16x^2+24x-9$with the standard form $f\left(x\right)=a{x}^2+bx+c$we get

$$\begin{gathered} a=-16 \\ b=24 \\ c=-9 \end{gathered}$$

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

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{24}{2·(-16)} \\ x=-\frac{3}{-4} \\ x=0.75 \end{gathered}$$

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

$$\begin{gathered} f(0.75)=-16·{(0.75)}^2+24·0.75-9 \\ f(0.75)=-9+18-9 \\ f(0.75)=0 \end{gathered}$$

So the vertex of the parabola is $(0.75,0)$.

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

$$\begin{gathered} f\left(0\right)=-16·0^2+24·0-9 \\ f\left(0\right)=-0+0-9 \\ f\left(0\right)=-9 \end{gathered}$$

Thus they intercept is $(0,-9)$.

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

$\begin{array}{rcl}f\left(x\right)& =& -16x^2+24x-9\\ 0& =& -16x^2+24x-9\end{array}$

Factoring the perfect square trinomial we get

$\begin{array}{rcl}0& =& -(16x^2-24x+9)\\ 0& =& -({\left(4x\right)}^2-2·4x·3+3^2)\\ 0& =& -{(4x-3)}^2\end{array}$

So the solution is

$\begin{array}{rcl}4x-3& =& 0\\ 4x& =& 3\\ x& =& 0.75\end{array}$

So the xintercept is$(0.75,0)$.

We have y intercept as $(0,-9)$.

The axis of symmetry is$x=0.75$.
It means that the intercept is at a distance of$0.75$ units to the left 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 right will also lie on the graph. This point will be $(1.5,-9)$.

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