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.

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.

Answer

Explanation:

Step1: Identify the vertex $(h, k)$ from the graph

The vertex is the lowest point of the parabola, located at $(6, 1)$.

Step2: Determine the horizontal and vertical shifts

The parent function $f(x) = x^2$ has a vertex at $(0, 0)$. The graph is shifted right by $h = 6$ and up by $k = 1$.

Step3: Use a point to find the scale factor $a$

Pick a point on the graph, such as $(5, 2)$. Substitute into $g(x) = a(x - h)^2 + k$: $$2 = a(5 - 6)^2 + 1$$

Step4: Solve for the value of $a$

$$2 = a(-1)^2 + 1 \implies 2 = a + 1 \implies a = 1$$

Step5: Write the final function rule

Substitute $a = 1$, $h = 6$, and $k = 1$ into the vertex form equation. $$g(x) = 1(x - 6)^2 + 1$$

Answer:

$g(x) = (x - 6)^2 + 1$