Question

1. use the equation $y = 4(0.67)^t$ to answer the following questions

a. determine the initial value

b. determine the rate of change

c. does this represent exponential growth or exponential decay?

d. evaluate the equation when $t = 2$

Explanation:

Part a:

Step1: Recall exponential form

The general exponential function is \( y = a(b)^t \), where \( a \) is the initial value (when \( t = 0 \)).

Step2: Identify \( a \) in given equation

In \( y = 4(0.67)^t \), when \( t = 0 \), \( y = 4(0.67)^0 = 4\times1 = 4 \). So the initial value is determined by the coefficient of the exponential term.

Step1: Recall exponential rate formula

For \( y = a(b)^t \), the base \( b \) relates to the rate. The rate of change (decay or growth factor) is \( b \), and the percentage rate can be found from \( b = 1 + r \) (growth) or \( b = 1 - r \) (decay).

Step2: Identify \( b \) and find rate

Here, \( b = 0.67 \). Since \( 0.67=1 - r \), solving for \( r \): \( r = 1 - 0.67 = 0.33 \) or \( 33\% \) decay rate. The rate of change factor is \( 0.67 \), and the percentage rate of decay is \( 33\% \).

Step1: Recall growth/decay condition

For \( y = a(b)^t \), if \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay.

Step2: Check \( b \) value

In \( y = 4(0.67)^t \), \( b = 0.67 \), and \( 0 < 0.67 < 1 \). So it represents exponential decay.

Answer:

The initial value is \( 4 \).

Part b: