Answer
$$
f(x)=-\frac{1}{2}x^{2}-x-\frac{7}{2}
$$
There are no x-intercepts.
The $y$-intercept is $-\frac{7}{2}$..
Vertex parabola is $\left(-1 ,-3\right)$.
The axis is $x=-1$ , the vertical line through the vertex.
Plotting the vertex, the $y$-intercept,the $x$-intercepts. and the point $\left(-1 ,-3\right)$ gives the graph in Figure
Work Step by Step
$$
f(x)=-\frac{1}{2}x^{2}-x-\frac{7}{2}
$$
The $x$-intercepts can be found by letting $f(x)=0$ to get
$$
f(x)=-\frac{1}{2}x^{2}-x-\frac{7}{2} =-x^{2}-2x-7=0
$$
This does not appear to factor, so we’ll try the quadratic formula.
$$
\begin{split}
x_{1,\:2}&=\frac{-\left(-2\right)\pm \sqrt{\left(-2\right)^2-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)} \\
&=\frac{2\pm \sqrt{-24}}{-2} \\
\end{split}
$$
Since the radical is negative, there are no
x-intercepts.
To find the $y$-intercept , set $x=0 $
$$
f(0)=-\frac{1}{2}(0)^{2}-(0)-\frac{7}{2}=-\frac{7}{2}
$$
So the $y$-intercept is $-\frac{7}{2}$.
The $x$-coordinate of the vertex is :
$$
x=\frac{-b}{2a}=-\frac{-1}{2(-\frac{1}{2})}=\frac{-1}{1}=-1
$$
Substituting this into the equation gives
$$
f(-1)=-\frac{1}{2}(-1)^{2}-(-1)-\frac{7}{2}=-\frac{1}{2}+1-\frac{7}{2}=-3
$$
The vertex is $\left(-1 ,-3\right)$.
The axis is $x=-1 $ , the vertical line through the vertex.
Plotting the vertex, the $y$-intercept,the $x$-intercepts. and the point $\left(-1 ,-3\right)$.. gives the graph in Figure