(Warm-Up: Write an equation that you can use to determine how many teams are playing in each round of the NCAA tournament. )

(Objective: Students will be able to write an exponential equation. )How To Write An Exponential Function When Given Two Points

1. Find the value of a. This is usually given to use as point (0,a)

2. Use the second point and plug in for x and y

3. Solve for b

·

If the exponent is positive, use the found in the MATH menu

·

If the exponent is negative, flip the b and cross multiply. Then use the found in the MATH menu

Example1: Write an exponential equation for a graph that passes through the points (0,3) and (3,192)

Example 2: Write an exponential equation for a graph that passes through the points (0,9) and (-2,81).

Example 3: A biologist is researching a newly-discovered species of bacteria. At 9am, he puts one hundred bacteria into what he has determined to be a favorable growth medium. At 3 pm, he measures 450 bacteria. Assuming exponential growth and that x is the number of hours since 9 am, predict the amount of bacteria that will be present at 5pm.

Example 4: At the beginning of a study there are 50 grams of a substance present. After 17 days, there are 38.7 grams remaining. How much of the substance will be present after 40 days? Assuming the substance decays exponentially, what is the rate of decay?

Count how many m&m’s are in your cup. Record this number as the first data entry. Shake the m & m’s and carefully pour them onto the paper plate. Remove (and eat!) the m & m’s that landed with the “m” face up. Count the remaining m&m’s and record the number. Repeat this procedure until there are less than 5 m&m’s remaining.

Trials

m&m’s

0

1

2

3

4

1. Create a table from your data.

On your graph paper, answer the following questions.

2. Graph your data on graph paper (Do not connect the dots!!).

3. What type of function does your scatter plot look like?

4. Name the y – intercept. What does this represent?

5. Write an exponential equation that models this data.

6. What is the domain of this function?

7. What is the range of this function?

8. Would you say that this is a situation that models exponential growth or exponential decay? Explain.

9. Use your equation to predict the number of M&M’s after the 12th trial.

10. Explain why your answer in #9 does not make sense in the “real world.”

11. After you shook your cup, approximate the percentage of m&m’s that landed with the “M” up.

12. Explain the meaning of a and C in your equation.

13. Finish the following sentence: In a perfect world, the value of “a” should be equal to __________ and the value of “b” should be equal to __________. Explain why you wrote what you did in the blanks.

Enter your data into L1 and L2 in your graphing calculator. The shape of our scatter plot appears to be exponential. We are now going to learn how to use the graphing calculator to find Exponential Regression (i.e. the exponential curve that best fits our data)

14. What regression formula did you come up with?

15. Graph the regression curve on the same axes as your scatter plot in question 2.

16. How does the regression equation compare to your original equation in #5?

17. How close are your calculated value of a & b to the ideal values from question #13? Explain why some differences may occur.

Name:

BACTERIA GROWTH

Time

Bacteria

8 A.M.

100

9 A.M.

300

12 P.M.

The number of bacteria in a colony is growing exponentially. Use the information to the right to answer the questions.

1. Write an equation that models this situation.

2. How many bacteria will be present at 12 pm?

3. Graph the values from the table on graph paper.

4. What type of graph does the data form?

5. State the domain and range.

6. Name the y – intercept. What does this represent?

Use the following information to answer the questions below.

African Black Rhino Population

Year

Population (in thousands)

1960

100

1980

15

1991

3.5

1992

2.4

7. Make a scatterplot for the given data.

8. Write an exponential equation that could model this data.

9. Predict the black rhino population for the years 1998 and 2004.

10. Should scientists be concerned about this decrease in population? Explain why or why not.

5. Solve for x

a.

b.

6. Graph each equation and explain if each equation is growth or decay.

a.

b.

7. Write an exponential function that passes through both points.

a. (0,2) & (2,72)

b. (0,4) & (-5,97

x

x

€

8x = 24x−2

 

8

x

=2

4x-2

€

6x+1 = 2162x−4

 

6

x+1

=216

2x-4

€

y = (3)(2)x

 

y=(3)(2)

x

€

y = (5)(14)x

 

y=(5)(

1

4

)

x