Question

which of the following is the correct equation for the given graph? f(x)=(x - 2)^2 - 1 f(x)=(x + 2)^2 - 1 f(x)=-(x - 2)^2 - 1 f(x)=-(x + 2)^2 - 1

Explanation:

Step1: Recall vertex - form of parabola

The vertex - form of a parabola is $y=a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. If $a>0$, the parabola opens upward, and if $a < 0$, the parabola opens downward.

Step2: Analyze the vertex of the given graph

The vertex of the parabola in the graph is at the point $(2, - 1)$. So, $h = 2$ and $k=-1$.

Step3: Determine the sign of $a$

The parabola opens downward, so $a=-1$.

Step4: Write the equation

Substitute $a=-1$, $h = 2$, and $k = - 1$ into the vertex - form $y=a(x - h)^2+k$. We get $y=-(x - 2)^2-1$.

Answer:

$f(x)=-(x - 2)^2-1$