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

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

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

Answer

Explanation:

Step1: Identify the vertex of the parabola

From the graph, the lowest point (vertex) is at $(4, 0)$.

Step2: Determine the horizontal and vertical shifts

The parent function $f(x) = x^{2}$ has a vertex at $(0, 0)$. The graph is shifted $4$ units right and $0$ units up, so $h = 4$ and $k = 0$.

Step3: Determine the leading coefficient $a$

Observe another point on the graph, such as $(5, 1)$. Substitute into $g(x) = a(x - 4)^{2} + 0$: $$1 = a(5 - 4)^{2} \implies 1 = a(1)^{2} \implies a = 1$$

Step4: Write the final function rule

Substitute $a = 1$, $h = 4$, and $k = 0$ into the vertex form $g(x) = a(x - h)^{2} + k$. $$g(x) = 1(x - 4)^{2} + 0$$

Answer:

$g(x) = (x - 4)^{2}$