Question

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease.
$y = 7000(1.97)^x$

Explanation:

Step 1: Recall the exponential growth formula

The general form of an exponential growth function is \( y = a(1 + r)^x \), where:

• \( a \) is the initial amount,
• \( r \) is the growth rate (as a decimal),
• \( x \) is the time variable.

Step 2: Compare with the given function

The given function is \( y = 7000(1.97)^x \). Comparing this with the general form \( y = a(1 + r)^x \), we can see that \( 1 + r = 1.97 \).

Step 3: Solve for \( r \)

To find \( r \), we subtract 1 from both sides of the equation \( 1 + r = 1.97 \):
\[
r = 1.97 - 1 = 0.97
\]

Step 4: Convert \( r \) to a percentage

To convert the decimal \( r = 0.97 \) to a percentage, we multiply by 100:
\[
\text{Percentage rate} = 0.97 \times 100 = 97\%
\]

Since the base \( 1.97 \) is greater than 1, the function represents exponential growth with a percentage rate of increase of \( 97\% \).

Answer:

The function represents exponential growth with a percentage rate of increase of \( 97\% \).