Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.
$y = 88(0.977)^t$

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 percentage rate of change is \( |b - 1| \times 100\% \).

Step2: Analyze the given function

Given \( y = 88(0.977)^x \), the base \( b = 0.977 \). Since \( 0 < 0.977 < 1 \), this represents decay.

Step3: Calculate the percentage rate of decrease

To find the percentage rate of decrease, we calculate \( (1 - 0.977) \times 100\% \).
\( 1 - 0.977 = 0.023 \)
\( 0.023 \times 100\% = 2.3\% \)

Answer:

The function represents decay with a percentage rate of decrease of \( 2.3\% \).