Question

omplete the table to investigate dilations of exponential functions.
which function represents a horizontal compression of an exponential function?

$x$	$2^x$	$3\cdot 2^x$	$2^{3x}$
$-1$	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{1}{8}$
$0$	$1$	$3$	$1$
$1$	$2$	$6$	$8$
$2$	$4$	$12$	$64$

$y = 3\cdot 2^x$
$y = 2^x$
$y = 2^{3x}$

Explanation:

Step1: Recall horizontal compression rule

For a function \( y = f(kx) \), if \( |k|>1 \), it's a horizontal compression of \( y = f(x) \).

Step2: Analyze each option

• \( y = 3\cdot2^{x} \): This is a vertical stretch (by factor 3) of \( y = 2^{x} \), not horizontal compression.
• \( y = 2^{x} \): This is the parent function, no compression.
• \( y = 2^{3x} \): Here, \( f(x)=2^{x} \), and \( k = 3>1 \), so it's a horizontal compression of \( y = 2^{x} \).

Answer:

\( y = 2^{3x} \) (the third option: \( y = 2^{3x} \))