Question

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

( y = 30(1.003)^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. 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

For the function \( y = 30(1.003)^x \), the base \( b = 1.003 \). Since \( 1.003>1 \), this represents growth.

Step3: Calculate the growth rate

Using the formula \( b = 1 + r \), we substitute \( b = 1.003 \):
\( 1.003=1 + r \)
Subtract 1 from both sides:
\( r=1.003 - 1=0.003 \)
To convert this to a percentage, multiply by 100:
\( r = 0.003\times100 = 0.3\% \)

Answer:

The change represents growth with a percentage rate of increase of \( 0.3\% \).