Question

the function ( f ) is given by ( f(x) = 2^x ), and the function ( g ) is given by ( g(x) = \frac{f(x)}{8} ). for which of the following transformations is the graph of ( g ) the image of the graph of ( f )?
a a horizontal translation to the left 3 units
b a horizontal translation to the right 3 units
c a vertical translation up ( \frac{1}{8} ) unit

Explanation:

Step1: Recall Function Transformation Rules

For a function \( y = f(x) \), a vertical translation is of the form \( y = f(x)+k \) (up if \( k>0 \), down if \( k < 0 \)), and a horizontal translation is of the form \( y=f(x - h) \) (right if \( h>0 \), left if \( h < 0 \)). Also, \( \frac{1}{a}f(x) \) represents a vertical scaling or a vertical translation? Wait, no, \( \frac{1}{8}f(x)=\frac{1}{2^{3}}f(x)=2^{- 3}f(x) \). Wait, \( f(x)=2^{x} \), so \( g(x)=\frac{f(x)}{8}=\frac{2^{x}}{8}=2^{x - 3} \) (since \( \frac{2^{x}}{2^{3}}=2^{x-3} \)).

Step2: Identify the Transformation

The function \( g(x)=2^{x - 3}=f(x - 3) \) (since \( f(x)=2^{x} \), so \( f(x - 3)=2^{x - 3} \)). By the horizontal translation rule \( y = f(x - h) \) is a translation of \( y = f(x) \) to the right by \( h \) units (when \( h>0 \)). Here \( h = 3 \), so it's a horizontal translation to the right by 3 units. Wait, but let's check the options. Option B is "A horizontal translation to the right 3 units", option A is left 3, option C is vertical up \( \frac{1}{8} \). Wait, but \( \frac{f(x)}{8}=2^{x}/8 \), which is also equal to \( 2^{x-3} \), so it's a horizontal translation to the right 3 units (since \( f(x - 3)=2^{x - 3}=\frac{2^{x}}{8}=\frac{f(x)}{8} \)). So the transformation is a horizontal translation to the right 3 units.

Answer:

B. A horizontal translation to the right 3 units