Question

1
compare the graph of each ( g(x) ) function listed below to the graph of ( f(x) = 2^x ).

( f(x) \to g(x) )	translate 2 units left	translate 2 units up	translate 2 units down	translate 2 units right
( g(x) = 2^{x - 2} )	( circ )	( circ )	( circ )	( circ )
( g(x) = 2^x - 2 )	( circ )	( circ )	( circ )	( circ )
( g(x) = 2^{x + 2} )	( circ )	( circ )	( circ )	( circ )

Explanation:

To solve this, we use the rules of function transformations:

• For vertical shifts: \( g(x) = f(x) + k \) shifts \( f(x) \) up by \( k \) (if \( k>0 \)) or down by \( |k| \) (if \( k<0 \)).
• For horizontal shifts: \( g(x) = f(x - h) \) shifts \( f(x) \) right by \( h \) (if \( h>0 \)) or left by \( |h| \) (if \( h<0 \)).

For \( g(x) = 2^x + 2 \):

This is \( f(x) + 2 \) (where \( f(x) = 2^x \)). By vertical shift rule, adding 2 shifts the graph 2 units up.

For \( g(x) = 2^{x - 2} \):

This is \( f(x - 2) \) (where \( f(x) = 2^x \)). By horizontal shift rule, \( h = 2>0 \), so it shifts the graph 2 units right.

For \( g(x) = 2^x - 2 \):

This is \( f(x) - 2 \) (where \( f(x) = 2^x \)). By vertical shift rule, subtracting 2 shifts the graph 2 units down.

For \( g(x) = 2^{x + 2} \):

Rewrite as \( 2^{(x - (-2))} \), so this is \( f(x - (-2)) \) (where \( f(x) = 2^x \)). By horizontal shift rule, \( h = -2 \), so it shifts the graph 2 units left (since \( h<0 \)).

Final Table (with selections):

\( f(x) \to g(x) \)	Translate 2 units left	Translate 2 units up	Translate 2 units down	Translate 2 units right
\( g(x) = 2^{x - 2} \)				\(\boldsymbol{\circ}\)
\( g(x) = 2^x - 2 \)			\(\boldsymbol{\circ}\)
\( g(x) = 2^{x + 2} \)	\(\boldsymbol{\circ}\)

(Note: In the table, mark the correct column with \(\boldsymbol{\circ}\) for each \( g(x) \).)

Answer:

To solve this, we use the rules of function transformations:

• For vertical shifts: \( g(x) = f(x) + k \) shifts \( f(x) \) up by \( k \) (if \( k>0 \)) or down by \( |k| \) (if \( k<0 \)).
• For horizontal shifts: \( g(x) = f(x - h) \) shifts \( f(x) \) right by \( h \) (if \( h>0 \)) or left by \( |h| \) (if \( h<0 \)).

For \( g(x) = 2^x + 2 \):

This is \( f(x) + 2 \) (where \( f(x) = 2^x \)). By vertical shift rule, adding 2 shifts the graph 2 units up.

For \( g(x) = 2^{x - 2} \):

This is \( f(x - 2) \) (where \( f(x) = 2^x \)). By horizontal shift rule, \( h = 2>0 \), so it shifts the graph 2 units right.

For \( g(x) = 2^x - 2 \):

This is \( f(x) - 2 \) (where \( f(x) = 2^x \)). By vertical shift rule, subtracting 2 shifts the graph 2 units down.

For \( g(x) = 2^{x + 2} \):

Rewrite as \( 2^{(x - (-2))} \), so this is \( f(x - (-2)) \) (where \( f(x) = 2^x \)). By horizontal shift rule, \( h = -2 \), so it shifts the graph 2 units left (since \( h<0 \)).

Final Table (with selections):

\( f(x) \to g(x) \)	Translate 2 units left	Translate 2 units up	Translate 2 units down	Translate 2 units right
\( g(x) = 2^{x - 2} \)				\(\boldsymbol{\circ}\)
\( g(x) = 2^x - 2 \)			\(\boldsymbol{\circ}\)
\( g(x) = 2^{x + 2} \)	\(\boldsymbol{\circ}\)

(Note: In the table, mark the correct column with \(\boldsymbol{\circ}\) for each \( g(x) \).)