Question

which equations matches the graph above?
$h(x) = -(x)^2 - 4$
$h(x) = (x + 4)^2 - 1$
$h(x) = (x + 1)^2 - 4$
$h(x) = -(x + 4)^2 + 1$

Explanation:

Step1: Recall vertex form of parabola

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

Step2: Identify vertex from graph

From the graph, the vertex (lowest point) is at $(-4, -1)$. So $h=-4$, $k=-1$.

Step3: Check parabola direction

The parabola opens upward, so $a>0$.

Step4: Match to options

Substitute $h=-4$, $k=-1$, $a=1$ into vertex form:
$h(x)=(x-(-4))^2+(-1)=(x+4)^2-1$
This matches one of the options.

Answer:

$h(x)=(x+4)^2-1$