Question

write the equation of the parabola that has its vertex at (-5, -3) and passes through the point (-4, -5) in f(x)=a(x - h)^2 + k form.
the equation of the parabola in f(x)=a(x - h)^2 + k form is f(x)=□.

Explanation:

Step1: Identify h and k

The vertex - form of a parabola is $f(x)=a(x - h)^{2}+k$, where $(h,k)$ is the vertex. Given the vertex $(-5,-3)$, we have $h=-5$ and $k = - 3$. So the equation becomes $f(x)=a(x + 5)^{2}-3$.

Step2: Find the value of a

Since the parabola passes through the point $(-4,-5)$, substitute $x=-4$ and $y = - 5$ into the equation $f(x)=a(x + 5)^{2}-3$.
We get $-5=a(-4 + 5)^{2}-3$.
Simplify the right - hand side: $-5=a(1)^{2}-3$, which is $-5=a - 3$.
Add 3 to both sides of the equation: $a=-5 + 3=-2$.

Step3: Write the final equation

Substitute $a=-2$ into $f(x)=a(x + 5)^{2}-3$. The equation of the parabola is $f(x)=-2(x + 5)^{2}-3$.

Answer:

$f(x)=-2(x + 5)^{2}-3$