Question

exponential growth and decay worksheet

1. $y = 1200 \cdot (1 + 0.3)^t$

a. does this function represent exponential growth or exponential decay?
b. what is your initial value?
c. what is the rate of growth or rate of decay?

1. $y = 55 \cdot (1 - 0.02)^t$

a. does this function represent exponential growth or exponential decay?
b. what is your initial value?
c. what is the rate of growth or rate of decay?

1. $y = 100 \cdot (1.25)^t$

a. does this function represent exponential growth or exponential decay?
b. what is your initial value?
c. what is the rate of growth or rate of decay?

Explanation:

Problem 1: \( y = 1200 \cdot (1 + 0.3)^t \)

A. Growth or Decay?

Step 1: Analyze the base

The general form of an exponential function is \( y = a \cdot (1 + r)^t \) for growth (where \( r>0 \)) and \( y = a \cdot (1 - r)^t \) for decay (where \( r>0 \)). Here, the base is \( 1 + 0.3=1.3 \), which is greater than 1.

Step 2: Determine growth/decay

Since the base \( 1.3>1 \), this represents exponential growth.

Step 1: Recall initial value definition

The initial value of an exponential function \( y = a \cdot b^t \) is the value when \( t = 0 \). When \( t = 0 \), \( b^0=1 \), so \( y=a \).

Step 2: Identify \( a \)

In \( y = 1200 \cdot (1 + 0.3)^t \), \( a = 1200 \).

Step 1: Recall growth rate formula

For \( y = a \cdot (1 + r)^t \), \( r \) is the growth rate (in decimal). Here, \( 1 + r=1.3 \).

Step 2: Solve for \( r \)

Subtract 1 from both sides: \( r=1.3 - 1 = 0.3 \), or \( 30\% \) when converted to a percentage.

Answer:

Exponential Growth

B. Initial Value