Question

complete the table to investigate dilations of exponential functions.
which function represents a vertical stretch of an exponential function?
$y = 2^x$
$y = 2^{3x}$
$y = 3 \cdot 2^x$

Explanation:

Step1: Recall vertical stretch rule

For an exponential function \( y = a\cdot b^{x} \), a vertical stretch by a factor of \( a \) (where \( a>1 \)) occurs when we multiply the original function \( y = b^{x} \) by \( a \).

Step2: Analyze each function

• For \( y = 2^{x} \): This is the original exponential function with \( a = 1 \), so no vertical stretch.
• For \( y=2^{3x} \): This is a horizontal compression (since the exponent has a coefficient \( 3 \)), not a vertical stretch. The form is \( y = b^{kx} \) which affects the horizontal direction.
• For \( y = 3\cdot2^{x} \): Here, we are multiplying the original function \( y = 2^{x} \) by \( 3 \) (where \( 3>1 \)), which is a vertical stretch by a factor of \( 3 \) according to the vertical stretch rule for functions.

Answer:

\( y = 3\cdot2^{x} \)