Question

f(x) = -2(x + 3)^2 - 1; f(x) = -2(x + 3)^2 + 1; f(x) = 2(x + 3)^2 + 1; f(x) = 2(x - 3)^2 + 1

Explanation:

Step1: Recall vertex form

The vertex form of a parabola is $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex, and $a$ determines direction/wideness.

Step2: Analyze top graph

Vertex at $(-3,1)$, opens upward ($a>0$).
Matches $f(x)=2(x+3)^2+1$ (since $h=-3, k=1, a=2$).

Step3: Analyze middle graph

Vertex at $(-3,1)$, opens downward ($a<0$).
Matches $f(x)=-2(x+3)^2+1$ (since $h=-3, k=1, a=-2$).

Step4: Analyze bottom graph

Vertex at $(3,1)$, opens upward ($a>0$).
Matches $f(x)=2(x-3)^2+1$ (since $h=3, k=1, a=2$).

Answer:

Top graph: $f(x)=2(x+3)^2+1$
Middle graph: $f(x)=-2(x+3)^2+1$
Bottom graph: $f(x)=2(x-3)^2+1$