H. Algebra 2 Section 8.1 - Exponential Growth and Decay

How do graphs and equations reveal key features of exponential growth and decay functions?

Graph this function:

What are the key elements of this graph?

x-intercept: ______

y-intercept: ______

x / f(x)
-2
1
3

End behavior:Domain:

Asymptote:Range:

Parent Graph of Exponential Function:

Parts of an Exponential Function and how to find them:

Growth vs. Decay

When ______, then represents exponential ______.

x / f(x) = 3x
-2
-1
0
1
2

When ______, then represents exponential ______.

x /
-2
-1
0
1
2

What Happens if you Change the Base?

Compare the following graphs:

Conclusion:______.

Compare the following graphs:

Conclusion:______.

Classify each function as “growth” or “decay.” Graph each exponential function. Find the x-intercept (if it exists), y-intercept (if it exists), domain, range, and asymptote.

1. 2.

x-intercept: ______x-intercept: ______

y-intercept: ______y-intercept: ______

domain: ______domain: ______

range: ______range: ______

asymptote: ______asymptote: ______

3. 4.

x-intercept: ______x-intercept: ______

y-intercept: ______y-intercept: ______

domain: ______domain: ______

range: ______range: ______

asymptote: ______asymptote: ______

5. 6.

x-intercept: ______x-intercept: ______

y-intercept: ______y-intercept: ______

domain: ______domain: ______

range: ______range: ______

asymptote: ______asymptote: ______

Algebra 2 8.1: Exponential Growth and Decay (Word Problems)

Exponential GROWTH Problems:

Example 1: Bacteria Growth. eww.

Bacteria are very small single celled organisms that live almost everywhere on Earth. Most bacteria are not harmful to humans, and some are helpful, like bacteria in yogurt.

Bacteria reproduce by dividing. The total number of bacteria is called its “population.” When each bacterium divides, the population doubles.

Complete the following table based on bacteria growth.

Time (hours) / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / x
Population / 25 / 50 / 100

Analysis:

  1. Predict the population after 12 hours: ______
  1. Predict the population after 24 hours: ______

The population after x hours can be written as an ______

Example 2: Human Population Growth. Better than bacteria.

The Population of the United States was 321,442,019 on July 4, 2015

and is projected to grow at a rate of about 8% per decade.

Write the function for the population x decades after 2015:

Prediction for 2025:

Prediction for 2040:

Write a function, and solve each problem.

1.An investment of $75,000 increases at a rate of 12.5% per year. What is the value of the investment after 30 years?

2.A new home that sells for $350,000 appreciates 2% each year. What is the value of the home after 7 years?

3.In 2009, there were 1570 bears in a wildlife refuge. In 2010, the population had increased to approximately 1884 bears. If this trend continues and the bear population is increasing exponentially, how many bears will there be in 2010?

4.Suppose you deposit $1500 in a savings account that pays interest at an annual rate of 6%. No money is added or withdrawn from the account.

  1. How much will be in the account after 5 years?
  1. How much will be in the account after 20 years?
  1. How many years will it take for the account to contain $2500?
  1. How many years will it take for the account to contain $4000?

Money Growth. nice!

Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be part of the principal amount is called compounding (i.e., interest is compounded).

Compound Interest Formula:

The total amount of an investment earning compound interest:

P = principal (initial deposit)

r = annual interest rate

n = number of times the interest is compounded per year

t = time in years

Example 3: Find the final amount for a $1000 investment at 6% interest compounded annually for 20 years.

Practice:

4. Find final amount of a $100 investment after 10 years at 5% interest compounded annually.

5. Find the final amount of a $2500 investment after 15 years at 3.2% interest compounded quarterly.

6. Find the final amount of a $3600 investment after 6 years at 1.5% interest compounded daily.

Continuous Compound Interest Formula:

The total amount of an investment earning compound interest:

P = principal (initial deposit)

r = annual interest rate

t = time in years

Practice:

7. Find final amount of a $100 investment after 10 years at 5% interest compounded continuously.

8. Find the final amount of a $2500 investment after 15 years at 3.4% interest compounded continuously.

9. A student wants to save $8000 for college in five years. How much should she put into an account that earns 5.2% annual interest compounded continuously?

Exponential DECAY Problems:

Example 1: Biological Decay.

Let us say that there is 100 mg of caffeine per cup. Recall that 13% of the caffeine

in our bodies is eliminated each hour.

Write the function for the caffeine level x hours after peak level:

Prediction after 1 hour:

Prediction after 4 hours:

Write a function, and solve each situation.

1.A dye is injected into the pancreas during a certain medical procedure. A physician injects .3 grams of the dye, and a healthy pancreas will secrete (eliminate) 4% of the dye each minute. Predict the amount of dye remaining in a healthy pancreas 30 minutes after the injection.

  1. Your friend drops a rubber ball from 4 ft. You notice that its rebound is 32.5 in. on the first bounce and 22 in. on the second bounce.

What exponential function would be a good model for the height of the ball?

How high will the ball bounce on the fourth bounce?

3.A new truck that sells for $29,000 depreciates 12% each year. What is the value of the truck after 7 years?

4.A new laptop computer costs $1500. The value of the computer decreases by 18% each year.

How much is the computer worth after 2 years?

Half-Life. sounds a little depressing.

Half-Life: the time it takes for a radioactive substance to decay to 50% of its original mass.

Example 2: You have 23 grams of a radioactive substance with a half-life of 1350 years. Estimate how much of the radioactive substance will remain after 1000 years? 5000 years? 50,000 years?

5.When a plant or animal dies, it stops acquiring carbon-14 from the atmosphere. Carbon-14 decays over time with a half-life of 5730 years. What percent of the original carbon-14 remains in a sample after 2500 years? 5,000 years? 10,000 years?

6.Fluorine-21 has a half-life of approximately 5 seconds. What fraction of the original nuclei would remain after 1 minute?

7.The half-life of chromium-51 is 28 days. If the sample contained 510 grams, how much was present 168 days ago?

8. The isotope Sr-85 is used in bone scans. It has a half-life of 64.9 days. Write the exponential decay function for an 8-mg sample. Find the amount remaining after 100 days.