# Identify each of the following functions as exponential growth or decay. Then give the rate of growth or decay as a percent. y=a(3/2)^t

Question
Exponential growth and decay
Identify each of the following functions as exponential growth or decay. Then give the rate of growth or decay as a percent. $$\displaystyle{y}={a}{\left(\frac{{3}}{{2}}\right)}^{{t}}$$

2021-01-18
$$\displaystyle{y}={a}{\left(\frac{{3}}{{2}}\right)}^{{t}}$$
This is a growth function
Rate of decay is 1.5-1=0.5
That is 50%
Consider an exponential function of the form: $$\displaystyle{y}={b}\cdot{a}^{{x}}$$
If a>1, then it is exponential growth function with growth rate a-1
If a<1, then it is exponential decay function with decay rate 1-a

### Relevant Questions

Identify each of the following functions as exponential growth or decay. Then give the rate of growth or decay as a percent. $$\displaystyle{y}={a}{\left(\frac{{5}}{{4}}\right)}^{{t}}$$
Determine whether the function represents exponential growth or exponential decay. Then identify the percent rate of change. y=5(0.6)^t
Determine whether the function represents exponential growth or exponential decay. Then identify the percent rate of change. y=10(1.07)^t
The close connection between logarithm and exponential functions is used often by statisticians as they analyze patterns in data where the numbers range from very small to very large values. For example, the following table shows values that might occur as a bacteria population grows according to the exponential function P(t)=50(2t):
Time t (in hours)012345678 Population P(t)501002004008001,6003,2006,40012,800
a. Complete another row of the table with values log (population) and identify the familiar function pattern illustrated by values in that row.
b. Use your calculator to find log 2 and see how that value relates to the pattern you found in the log P(t) row of the data table.
c. Suppose that you had a different set of experimental data that you suspected was an example of exponential growth or decay, and you produced a similar “third row” with values equal to the logarithms of the population data.
How could you use the pattern in that “third row” to figure out the actual rule for the exponential growth or decay model?
Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change.
$$\displaystyle{y}={3}{\left({1.88}\right)}^{{t}}$$
Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. y=3(1.88)ty=3(1.88)^t
$$\displaystyle{a}{)}{y}={2}^{{{0.1}{x}}}$$
$$\displaystyle{b}{)}{y}={3}^{{{1.4}{x}}}$$
$$\displaystyle{c}{)}{y}={4}{e}^{{-{5}{x}}}$$
$$\displaystyle{d}{)}{y}={0.8}^{{{3}{x}}}$$
Identify the function as exponential growth or exponential decay.Then identify the growth or decay factor. $$\displaystyle{y}={5.2}\cdot{3}^{{x}}$$