Answer
$f(x)=-3(x+1)^2+5$ or $f(x)=-3x^2-6x+2$.
Work Step by Step
To find a quadratic function $f$ that has the assigned properties, first we need to write down a quadratic function's standard form:
$f(x)=a(x-h)^2+k$
As we know, the coordinates of the vertex describe the minimum/maximum value of that quadratic function and taking the form of $(h,k)$. Therefore, we can now rewrite our function as:
$f(x)=a[x-(-1)]^2+5$
$f(x)=a(x+1)^2+5$
In this question, they also give us the position of another point that belongs to the parabola. We can substitute the value of the coordinates to the function to find $a$.
$f(x)=a(x+1)^2+5$
$-7=a(-3+1)^2+5$
$-7=a(-2)^2+5$
$-7=4a+5$
$4a=-12$
$a=-3$
In conclusion, the desired function is $f(x)=-3(x+1)^2+5$ or $f(x)=-3x^2-6x+2$.
