Question

given ( f(x) = 0.4x ) is transformed to ( g(x) = 0.4(x + 8) ), what transformation occurs?
vertical shift of 8 units up
vertical shift of 8 units down
horizontal shift of 8 units to the right
horizontal shift of 8 units to the left

Explanation:

Step1: Recall function transformation rules

For a function \( y = f(x) \), the transformation \( y = f(x + h) \) represents a horizontal shift. If \( h>0 \), it is a shift to the left by \( h \) units; if \( h < 0 \), it is a shift to the right by \(|h|\) units. Vertical shifts are of the form \( y=f(x)+k \) (up if \( k>0 \), down if \( k < 0 \)).

Step2: Analyze the given functions

We have \( f(x)=0.4x \) and \( g(x) = 0.4(x + B) \). Comparing with the transformation rule \( y=f(x + h) \), here \( h = B \) and the form is a horizontal shift (since the change is inside the function argument with \( x \)). If we assume \( B>0 \) (from the context of the options, likely a horizontal shift left by \( B \) units, but looking at the options, the last option which is "Horizontal shift of \( B \) units to the left" (assuming the text in the option is about horizontal shift left) matches. The other options are vertical shifts (involving adding/subtracting outside the function), but here the change is inside the function, so it's a horizontal shift.

Answer:

Horizontal shift of \( B \) units to the left (assuming the option with horizontal shift left is the correct one, based on the function transformation rules)