Question

what kind of transformation converts the graph of $f(x) = 9x - 3$ into the graph of $g(x) = x - 3$?
horizontal shrink
vertical shrink
horizontal stretch
vertical stretch

Explanation:

Step1: Recall transformation rules

For a linear function \( y = a x + b \), a vertical stretch or shrink is given by \( y = k(ax + b) \) (vertical) and horizontal by \( y = a(kx)+b \) (horizontal). Here, \( f(x)=9x - 3 \), \( g(x)=x - 3 \). We can write \( g(x)=\frac{1}{9}(9x - 3) \), which is a vertical shrink (since \( 0<\frac{1}{9}<1 \)) of \( f(x) \).

Step2: Eliminate other options

Horizontal transformations involve \( x \)-coefficient changes (e.g., \( f(kx) \)), but here the \( x \)-coefficient in \( f(x) \) is 9, in \( g(x) \) is 1, and the transformation is on the whole function (multiplying by \( \frac{1}{9} \)), so vertical. Stretch would be \( |k|>1 \), shrink \( 0<|k|<1 \), so vertical shrink.

Answer:

vertical shrink