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

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

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

Answer

Explanation:

Step1: Identify the vertex of the parabola

The vertex $(h, k)$ is the lowest point on the graph, located at $(-7, 5)$.

Step2: Determine the horizontal and vertical shifts

The parent function $f(x) = x^2$ has a vertex at $(0, 0)$. The graph is shifted left by $7$ units ($h = -7$) and up by $5$ units ($k = 5$).

Step3: Determine the leading coefficient $a$

Observe a point one unit away from the vertex, such as $(-6, 6)$. Since the $y$-value increases by $1$ when the $x$-value increases by $1$, $a = 1$.

Step4: Write the function in vertex form

Substitute $a = 1$, $h = -7$, and $k = 5$ into the vertex form $g(x) = a(x - h)^2 + k$. $$g(x) = 1(x - (-7))^2 + 5$$

Step5: Simplify the expression

Simplify the signs inside the parentheses. $$g(x) = (x + 7)^2 + 5$$

Answer:

g(x) = (x + 7)^2 + 5