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

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

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

Answer

Explanation:

Step1: Identify the vertex of the parabola

The vertex $(h, k)$ is the highest or lowest point. From the graph, the vertex is at $(7, -1)$. $$h = 7, k = -1$$

Step2: Identify another point on the graph

The parabola passes through the point $(4, -2)$. $$(x, y) = (4, -2)$$

Step3: Use the vertex form equation

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

Step4: Solve for the leading coefficient $a$

Substitute the point $(4, -2)$ into the equation and solve for $a$. $$-2 = a(4 - 7)^2 - 1$$ $$-2 = a(-3)^2 - 1$$ $$-2 = 9a - 1$$ $$-1 = 9a$$ $$a = -\frac{1}{9}$$

Step5: Write the final function rule

Substitute $a$, $h$, and $k$ back into the vertex form. $$g(x) = -\frac{1}{9}(x - 7)^2 - 1$$

Answer:

g(x) = -\frac{1}{9}(x - 7)^2 - 1