Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.

$y = 3800(0.97)^x$

Explanation:

Step1: Recall exponential form

The general form of an exponential function is \( y = a(b)^x \), where \( a \) is the initial amount and \( b \) is the base. If \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay.
Here, \( b = 0.97 \), and \( 0 < 0.97 < 1 \), so it's decay.

Step2: Find percentage rate

The formula for the rate of decay is \( r = 1 - b \) (expressed as a decimal), then convert to percentage.
\( r = 1 - 0.97 = 0.03 \)
To convert to percentage, multiply by 100: \( 0.03\times100 = 3\% \)

Answer:

The function represents decay with a percentage rate of decrease of \( 3\% \).