Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.\\(y = 570(1.01)^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, \( b \) is the base. If \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay. The growth/decay rate \( r \) is related to \( b \) by \( b=1 + r \) (for growth) or \( b = 1 - r \) (for decay).

Step2: Analyze the given function

Given \( y = 570(1.01)^x \), here \( b = 1.01 \). Since \( 1.01>1 \), this represents exponential growth.

Step3: Find the percentage rate

Using \( b = 1 + r \), substitute \( b = 1.01 \):
\( 1.01=1 + r \)
Subtract 1 from both sides: \( r=1.01 - 1=0.01 \)
To convert to a percentage, multiply by 100: \( 0.01\times100 = 1\% \)

Answer:

The function represents exponential growth with a percentage rate of increase of \( 1\% \).