Question

Write the function:

Answer 1

Notice that our function $f(x)$ is a parabola, and the vertex form of a parabola is:$f(x)=a(x-h)^{2} +k$ where (h,k) are the coordinates of its vertex.
The first thing we need to do is find the vertex (h,k) and to do that we are going to use the vertex formula: $h= \frac{-b}{2a}$, and then we are going to evaluate our function at $h$ to find $k$.
We know for our problem that $a=-5$, and $b=-60$, so lets replace those values in our vertex formula to find $h$:
$h= \frac{-b}{2a}$
$h= \frac{-(-60)}{2(-5)}$
$h= \frac{60}{-10}$
$h=-6$
Now, we are going to evaluate our function at -6 to find $k$:
$k=f(-6)=-5(-6)^{2} -60(-6)-181$
$k=-180+360-181$
$k=-1$

Now that we have our vertex, where $h=-6$ and $k=-1$; lets replace those values in our vertex form equation:
$f(x)=-5[x-(-6)]^{2} +(-1)$
$f(x)=-5(x+6)^{2} -1$

We can conclude that the vertex form of our function is $f(x)=-5(x+6)^{2} -1$, and its vertex is (-6,-1).