Question

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

\\( y = 410(0.58)^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, \( b = 0.58 \), and \( 0<0.58<1 \), so it's decay.

Step2: Calculate percentage rate of decrease

The formula for the rate of decrease is \( r=1 - b \) (expressed as a percentage).
So \( r = 1 - 0.58 = 0.42 \), which is \( 0.42\times100\% = 42\% \).

Answer:

The function represents decay with a percentage rate of decrease of \( 42\% \).