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. $f(x)=2^{x - h}$, for $h>0$, will shift the graph up.

Explanation:

Step1: Recall Function Transformation Rules

For a function \( y = f(x - h) \) (where \( h>0 \)), the graph of \( y = f(x) \) is shifted horizontally. If \( h>0 \), it's a shift to the right by \( h \) units; if \( h<0 \), it's a shift to the left. Here, our base function is \( f(x)=2^x \), and the transformed functions are \( f(x)=2^{x - 1} \) (so \( h = 1>0 \)) and \( f(x)=2^{x - 2} \) (so \( h = 2>0 \)).

Step2: Analyze Each Statement

• Left Shift?: A left shift would be for \( y = f(x+h) \) (or \( y = f(x - (-h)) \) with \( -h<0 \)). Our function is \( f(x)=2^{x - h} \) with \( h>0 \), so not a left shift. Eliminate the "shift left" statement.
• Down Shift?: Vertical shifts are of the form \( y = f(x)+k \) (up if \( k>0 \), down if \( k<0 \)). Our function has no vertical shift (it's a horizontal transformation), so eliminate "shift down".
• Right Shift?: Since our function is \( f(x)=2^{x - h} \) with \( h>0 \), by the horizontal shift rule, this is a shift of \( f(x)=2^x \) to the right by \( h \) units. For \( h = 1 \), \( f(x)=2^{x - 1} \) is \( f(x)=2^x \) shifted right by 1; for \( h = 2 \), \( f(x)=2^{x - 2} \) is shifted right by 2. This matches the "shift right" statement.
• Up Shift?: Similar to the down shift, there's no vertical shift component here (the exponent change is horizontal), so eliminate "shift up".

Answer:

\( f(x) = 2^{x - h} \), for \( h>0 \), will shift the graph right.