Question

this is a multi - part item. consider the transformation $f(x)=2^{x - h}$, for positive values of $h$. using graphing technology, graph $f(x)=2^{x}, f(x)=2^{x - 1}$, and $f(x)=2^{x - 2}$ and determine which statement is true. $f(x)=2^{x - h}$, for $h>0$. will shift the graph left. $f(x)=2^{x - h}$, for $h>0$, will shift the graph down. $f(x)=2^{x - h}$, for $h>0$, will shift the graph right.

Explanation:

Step1: Recall Transformations of Exponential Functions

For a function \( y = a^{x - h} \), the horizontal shift is determined by the value of \( h \). If \( h>0 \), the graph of \( y = a^x \) is shifted \( h \) units to the right. If \( h < 0 \), the graph is shifted \( |h| \) units to the left. Vertical shifts are of the form \( y=a^x + k \), where \( k>0 \) shifts up and \( k < 0 \) shifts down.

Step2: Analyze the Given Functions

We have the parent function \( f(x)=2^x \), and the transformed functions \( f(x)=2^{x - 1} \) (here \( h = 1>0 \)) and \( f(x)=2^{x - 2} \) (here \( h=2>0 \)).

• For \( f(x)=2^{x-1} \), comparing to \( f(x)=2^x \), when we substitute \( x \) with \( x - 1 \) in the parent function, the graph of \( 2^x \) is shifted 1 unit to the right.
• For \( f(x)=2^{x - 2} \), comparing to \( f(x)=2^x \), when we substitute \( x \) with \( x - 2 \) in the parent function, the graph of \( 2^x \) is shifted 2 units to the right.

Step3: Evaluate the Statements

• The first statement says "will shift the graph left" which is incorrect because \( h>0 \) in \( f(x)=2^{x - h} \) leads to right shift, not left.
• The second statement says "will shift the graph down" which is incorrect because there is no vertical shift term (the form is horizontal shift, not vertical).
• The third statement says "will shift the graph right" which matches our analysis of the horizontal shift for \( y = a^{x - h} \) with \( h>0 \).

Answer:

The true statement is: \( \boldsymbol{f(x)=2^{x - h},\text{ for }h > 0,\text{ will shift the graph right}} \)