Question

which sequence of transformations applied to the graph of $f(x) = 4^x$ will yield the graph of $g(x) = 4^{x+2} + 3$

• horizontal shift 2 units to the right; vertical shift 3 units down
• horizontal shift 3 units to the right; vertical shift 2 units down
• horizontal shift 2 units to the right; vertical shift 3 units up
• horizontal shift 2 units to the left; vertical shift 3 units up
• horizontal shift 2 units to the left; vertical shift 3 units down
• horizontal shift 3 units to the left; vertical shift 2 units up

Explanation:

Step1: Recall transformation rules

For a function \( y = f(x + h) \), if \( h>0 \), it's a horizontal shift left by \( h \) units; if \( h<0 \), shift right by \( |h| \) units. For \( y = f(x)+k \), if \( k>0 \), vertical shift up by \( k \) units; if \( k<0 \), shift down by \( |k| \) units.

Step2: Analyze \( g(x)=4^{x + 2}+3 \) from \( f(x)=4^{x} \)

• For the exponent part \( x+2 \) in \( 4^{x + 2} \), comparing to \( f(x)=4^{x} \), we have \( h = 2>0 \), so horizontal shift left by 2 units.
• For the \( +3 \) in \( 4^{x + 2}+3 \), comparing to \( f(x)=4^{x} \), we have \( k = 3>0 \), so vertical shift up by 3 units.

Answer:

D. Horizontal shift 2 units to the left; Vertical shift 3 units up