Question

what is the function ( g(x) ) created from ( f(x) = x^2 ) by moving the graph left 7 units, adding vertical compression by a factor of ( \frac{1}{6} ), and shifting the graph down 8 units?
a ( g(x) = 7(x + 8)^2 + \frac{1}{6} )
b ( g(x) = \frac{1}{6}(x - 7)^2 + 8 )
c ( g(x) = \frac{1}{6}(x + 8)^2 + 7 )
d ( g(x) = \frac{1}{6}(x + 7)^2 - 8 )

Explanation:

Step1: Shift left 7 units

Replace $x$ with $x+7$: $f_1(x)=(x+7)^2$

Step2: Vertical compression by $\frac{1}{6}$

Multiply by $\frac{1}{6}$: $f_2(x)=\frac{1}{6}(x+7)^2$

Step3: Shift down 8 units

Subtract 8: $g(x)=\frac{1}{6}(x+7)^2 - 8$

Answer:

D. $g(x)=\frac{1}{6}(x + 7)^2 - 8$