Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.\\( y = 2500(1.04)^x \\)

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( y = a(b)^x \), where \( a \) is the initial amount, and \( b \) determines growth or decay. If \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay.
Here, \( b = 1.04 \), and \( 1.04>1 \), so it's growth.

Step2: Find percentage rate

The growth rate \( r \) is related to \( b \) by \( b=1 + r \). So, \( 1.04=1 + r \), solving for \( r \), we get \( r = 1.04 - 1=0.04 \). To convert to percentage, multiply by 100: \( 0.04\times100 = 4\% \).

Answer:

The function represents exponential growth with a percentage rate of increase of \( 4\% \).