91Ó°ÊÓ

We have $f\left(x\right)=-x^2+2x-7$.

The parabola will open downwards here, because the leading coefficient is negative. Then by finding the axis of symmetry, vertex, the intercepts and a point with help of axis of symmetry, will help us to plot the points of the graph and get the required graph.

Axis of symmetry is a vertical line given by $x=\frac{-b}{2a}$, where $a,b,c$are the coefficients of the equation when compared with the standard form of the quadratic equation, i.e., $a{x}^2+bx+c$.

So the axis of symmetry comes out to be,
$$\begin{gathered} x=\frac{-b}{2a} \\ x=-\frac{2}{2(-1)} \\ x=\frac{-2}{-2} \\ x=1 \end{gathered}$$
Thus the equation of axis of symmetry is$x=1$.

Vertex is a point on the line of axis of symmetry. Here axis of symmetry is $x=1$.

We will get the vertex by putting$x=1$ in the equation. This gives us,
$$\begin{gathered} =-x^2+2x-7 \\ =-{\left(1\right)}^2+2\left(1\right)-7 \\ =-1+2-7 \\ =-8+2 \\ =-6 \end{gathered}$$
We get the vertex as$(1,-6)$.

To find $y-$intercept, we need to put $x=0$in the equation. This gives us,

$$\begin{gathered} =-x^2+2x-7 \\ =-{\left(0\right)}^2+2\left(0\right)-7 \\ =0+0-7 \\ =-7 \end{gathered}$$
The intercept comes out to be$(0,-7)$.

To find $x-$intercept, we need to put $f\left(x\right)=0$to solve for $x$.

We will here use discriminant formula, i.e.,$D=\frac{-bÂ\pm \sqrt{b^2-4ac}}{2a}$ where

$$\begin{gathered} a=-1 \\ b=2 \\ c=-7 \end{gathered}$$
$$\begin{gathered} D=\frac{-2Â\pm \sqrt{{\left(2\right)}^2-4(-1)(-7)}}{2(-1)} \\ D=\frac{-2Â\pm \sqrt{4-28}}{-2} \\ D=\frac{-2Â\pm \sqrt{-24}}{-2} \end{gathered}$$We know that square root of a negative number is not real, so we can conclude that the function has no $x-$intercept.

We have $y-$intercept as $(0,-7)$.

The axis of symmetry is $x=1$.

It means that the intercept is at a distance of $1$ unit to the left of axis of symmetry.
We can get a point on the graph using this information.
A point at the same distance, i.e. , $1$ unit, but to the right will also lie on the graph. This point will be$(2,-7)$

Plotting all the points we got and the axis of symmetry give us a graph. Joining these points will give us the graph as-