Question

determining the equation of a function from a graph
what is the equation of the translated function?
$f(x)=(x - 1)^2 - 5$
$f(x)=(x + 1)^2 + 5$
$f(x)=(x - 5)^2 - 1$
$f(x)=(x + 5)^2 + 1$
$f(x)=x^2$ (dashed curve)

Explanation:

Step1: Identify parent function vertex

The parent function is $f(x)=x^2$, vertex at $(0,0)$.

Step2: Locate translated function vertex

From the graph, the translated parabola's vertex is $(-1, 6)$.

Step3: Use vertex form of parabola

Vertex form: $y=(x-h)^2+k$, where $(h,k)$ is vertex. Substitute $h=-1, k=6$:
$y=(x-(-1))^2+6=(x+1)^2+6$

Step4: Match to options

This matches the second option.

Answer:

B. $f(x)=(x + 1)^2 + 6$