Question

write the equation of the transformation of g(x), if the parent function ( y = x^2 ) is compressed by a factor of ( \frac{2}{3} ) and shifted left 8.

Explanation:

Step1: Recall vertical compression rule

For a parent function \( y = f(x) \), a vertical compression by a factor of \( a \) (where \( 0 < a < 1 \)) gives \( y = a f(x) \). Here, the parent function is \( y = x^2 \) and the compression factor is \( \frac{2}{3} \), so after compression, the function becomes \( y=\frac{2}{3}x^2 \).

Step2: Recall horizontal shift rule

For a horizontal shift left by \( h \) units, the transformation of \( y = f(x) \) is \( y = f(x + h) \). Here, we shift left by 8 units, so we replace \( x \) with \( x + 8 \) in the compressed function.
Substituting \( x \) with \( x + 8 \) in \( y=\frac{2}{3}x^2 \), we get \( g(x)=\frac{2}{3}(x + 8)^2 \).

Answer:

\( g(x)=\frac{2}{3}(x + 8)^2 \)