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
d a vertical translation down ( \frac{1}{8} ) unit

Explanation:

Step1: Recall function transformation rules

For a function \( y = f(x) \), a vertical transformation of the form \( y=\frac{1}{k}f(x) \) (where \( k>0 \)) is a vertical compression or stretch. Also, \( y = f(x - h) \) is a horizontal translation (right if \( h>0 \), left if \( h < 0 \)), and \( y=f(x)+k \) is a vertical translation (up if \( k>0 \), down if \( k < 0 \)).

Given \( f(x)=2^{x} \) and \( g(x)=\frac{f(x)}{8} \). We can rewrite \( \frac{1}{8} \) as \( 2^{-3} \), so \( g(x)=\frac{2^{x}}{8}=2^{x - 3} \).

Step2: Identify the transformation

Comparing \( g(x)=2^{x - 3} \) with \( f(x)=2^{x} \), we use the horizontal translation rule \( y = f(x - h) \) (right by \( h \) units when \( h>0 \)). Here \( h = 3 \), so it is a horizontal translation to the right 3 units. Wait, no, wait: Wait, \( \frac{2^{x}}{8}=2^{x}\times2^{- 3}=2^{x - 3} \)? Wait, no, \( \frac{2^{x}}{8}=\frac{1}{8}\times2^{x}=2^{-3}\times2^{x}=2^{x-3} \)? Wait, no, \( a^{m}\times a^{n}=a^{m + n} \), so \( 2^{-3}\times2^{x}=2^{x-3} \). So the function \( g(x) \) is \( f(x - 3) \) scaled? Wait, no, \( \frac{f(x)}{8}=\frac{1}{8}f(x) \), which is a vertical compression? Wait, I made a mistake. Let's correct:

\( f(x)=2^{x} \), \( g(x)=\frac{f(x)}{8}=\frac{1}{8}\times2^{x} \). We can also write \( \frac{1}{8}=2^{-3} \), so \( g(x)=2^{x-3} \)? No, \( 2^{x-3}=2^{x}\times2^{-3}=\frac{2^{x}}{8} \), yes. So the transformation from \( f(x)=2^{x} \) to \( g(x)=2^{x - 3} \) is a horizontal translation to the right 3 units? Wait, no, the rule for horizontal translation is \( y = f(x - h) \) is a shift right by \( h \) units. So if \( g(x)=f(x - 3) \), then it's a shift right by 3 units. But let's check the options. Wait, the options are about horizontal/vertical translations. Wait, maybe I misread the function. Wait, the problem says \( g(x)=\frac{f(x)}{8} \), so \( g(x)=\frac{1}{8}\times2^{x} \). Alternatively, \( \frac{1}{8}=2^{-3} \), so \( g(x)=2^{x}\times2^{-3}=2^{x - 3} \), which is a horizontal translation of \( f(x)=2^{x} \) to the right by 3 units (since \( f(x - 3)=2^{x - 3} \)). So the transformation is a horizontal translation to the right 3 units, which is option B.

Wait, but let's check the vertical translation. If we consider \( g(x)=\frac{1}{8}f(x) \), that's a vertical compression by a factor of \( \frac{1}{8} \), but the options are about translation (not compression/stretch). Wait, maybe the problem has a typo, or I misread. Wait, the options are translation (horizontal or vertical shift). Wait, \( \frac{2^{x}}{8}=2^{x-3} \), so it's a horizontal shift right 3 units (since \( f(x - 3)=2^{x - 3} \)). So the correct option is B.

Answer:

B. A horizontal translation to the right 3 units