Question

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

( y = 5700(1.24)^x )

answer

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, \( b = 1.24 \), and \( 1.24>1 \), so it's growth.

Step2: Calculate percentage rate

The growth rate \( r \) is found by \( b=1 + r \). So, \( r=b - 1 \).
Substitute \( b = 1.24 \): \( r=1.24 - 1=0.24 \).
To convert to a percentage, multiply by 100: \( 0.24\times100 = 24\% \).

Answer:

This is an exponential growth function with a percentage rate of increase of \( 24\% \).