Question

let (g(x)) be a vertical shrink by a factor of (\frac{1}{3}) and a reflection in the x - axis, followed by a translation 6 units up from the parent function (f(x)=x^{2}). write a rule for (g(x)).

Explanation:

Step1: Apply vertical shrink

For a vertical shrink of the function $y = f(x)$ by a factor of $a$, the new - function is $y = af(x)$. Here, $a=\frac{1}{3}$ and $f(x)=x^{2}$, so the function becomes $y=\frac{1}{3}x^{2}$.

Step2: Apply reflection in the x - axis

For a reflection of the function $y = g(x)$ in the x - axis, the new function is $y=-g(x)$. Since $g(x)=\frac{1}{3}x^{2}$, the function becomes $y =-\frac{1}{3}x^{2}$.

Step3: Apply vertical translation

For a vertical translation of the function $y = h(x)$ by $k$ units up, the new function is $y=h(x)+k$. Here, $h(x)=-\frac{1}{3}x^{2}$ and $k = 6$, so the function $g(x)=-\frac{1}{3}x^{2}+6$.

Answer:

$g(x)=-\frac{1}{3}x^{2}+6$