Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.\\(y = 75(1.98)^{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 found by \( b=1 + r \) (for growth) or \( b = 1 - r \) (for decay).
Here, \( b = 1.98 \). Since \( 1.98>1 \), it's growth.

Step2: Calculate growth rate

Using \( b = 1 + r \), substitute \( b = 1.98 \):
\( 1.98=1 + r \)
Subtract 1 from both sides: \( r=1.98 - 1=0.98 \)
To convert to a percentage, multiply by 100: \( 0.98\times100 = 98\% \)

Answer:

The function represents growth with a percentage rate of increase of \( 98\% \).