Question

the graph shows $g(x)$, which is a translation of $f(x) = x^2$. write the function rule for $g(x)$.

(graph: a coordinate plane with a parabola opening upwards. the vertex is at (2, 0), and it passes through (0, 4).)

write your answer in the form $a(x - h)^2 + k$, where $a$, $h$, and $k$ are integers or simplified fractions.

$g(x) = \square$

Explanation:

Step1: Identify vertex of $g(x)$

The vertex of $g(x)$ is $(2, 0)$.

Step2: Recall vertex form of parabola

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

Step3: Substitute vertex values

Substitute $h=2$, $k=0$: $g(x)=a(x-2)^2+0$

Step4: Find $a$ using a point on $g(x)$

Use point $(0,4)$: $4=a(0-2)^2$ → $4=4a$ → $a=1$

Step5: Write final function rule

Substitute $a=1$ into the equation.

Answer:

$g(x)=(x-2)^2$