Question

question 7 (1 point)
describe the transformation of $f(x) = x^2$ represented by $g$. then graph each function.
$g(x) = -(4x)^2$

Explanation:

Step1: Analyze horizontal stretch/compression

For a function \( y = f(bx) \), if \( |b|>1 \), it's a horizontal compression by a factor of \( \frac{1}{|b|} \). Here, \( g(x)=-(4x)^2 = -f(4x) \), so horizontal compression by \( \frac{1}{4} \) (since \( b = 4 \)).

Step2: Analyze reflection

The negative sign in front of \( (4x)^2 \) means a reflection over the \( x \)-axis (because \( g(x)=-f(4x) \), reflecting \( f(4x) \) over \( x \)-axis).

Answer:

The transformation of \( f(x)=x^2 \) to \( g(x)=-(4x)^2 \) is a horizontal compression by a factor of \( \frac{1}{4} \) followed by a reflection over the \( x \)-axis. To graph: For \( f(x)=x^2 \), it's a parabola opening up with vertex at \( (0,0) \). For \( g(x)=-(4x)^2 \), first compress \( f(x) \) horizontally by \( \frac{1}{4} \) (making it narrower) then reflect over \( x \)-axis, so it opens down with vertex at \( (0,0) \) and narrower width.