Question

the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x).
write your answer in the form a(x − h)² + k, where a, h, and k are integers or simplified fractions.

g(x) =

Explanation:

Step1: Identify vertex of $g(x)$

Vertex is $(-3, -5)$

Step2: Recall vertex form formula

Vertex form: $g(x)=a(x-h)^2+k$, where $(h,k)$ is vertex

Step3: Substitute vertex values

$h=-3$, $k=-5$, so $g(x)=a(x-(-3))^2+(-5)=a(x+3)^2-5$

Step4: Solve for $a$ using x-intercept

Use $x=-2, g(x)=0$:
$0=a(-2+3)^2-5$
$0=a(1)^2-5$
$a=1$

Step5: Write final function rule

$g(x)=(x+3)^2-5$

Answer:

$g(x)=(x+3)^2-5$