Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.\\(y = 640(0.23)^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 \( b \) is the base. If \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay.
Here, \( b = 0.23 \), and \( 0 < 0.23 < 1 \), so it's decay.

Step2: Calculate percentage rate of decrease

The formula for the rate of decrease is \( r=1 - b \) (expressed as a decimal), then convert to percentage.
\( r = 1 - 0.23 = 0.77 \)
To get the percentage, multiply by 100: \( 0.77\times100 = 77\% \)

Answer:

The function represents decay with a percentage rate of decrease of \( 77\% \).