Daily Outcomes: I can evaluate, analyze, and graph exponential functions. Why might plotting the data on a graph be helpful in analyzing the data?

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Tobias Carter
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3 1 Exponential Functions Daily Outcomes: I can evaluate, analyze, and graph exponential functions Would the increase in water usage mirror the increase in population? Explain.

Sep 11 8:32 PM Algebraic functions: functions with values that are obtained by adding, subtracting, multiplying, or dividing constants and the independent variable or raising the independent variable

1 3 1 Exponential Functions Daily Outcomes: I can evaluate, analyze, and graph exponential functions Would the increase in water usage mirror the increase in population? Explain. Why might plotting the data on a graph be helpful in analyzing the data? What would a graph of the data look like in general? Can the trend in the graph go on continually for this situation? Sep 11 8:32 PM Algebraic functions: functions with values that are obtained by adding, subtracting, multiplying, or dividing constants and the independent variable or raising the independent variable to a rational power. Transcendental functions: exponential and logarithmic functions, ones that cannot be expressed in terms of algebraic operations Exponential functions: functions where the base is a constant and the exponent is the variable: y = 4 x Jul 29 6:50 PM 1

2 Jul 29 6:57 PM The inputs are rational numbers, exponents can be rational or irrational, all real numbers. f(x) = 4 x Oct 12 10:04 AM 2

$fraction y values 1. f(x) = 4 x 2.$

3 Jul 29 6:59 PM Example 1: Sketch the graph of each function. Steps. 1. find the y intercept 2. open x y chart and find two integer y values and two fraction y values 1. f(x) = 4 x 2. f(x) = 5 x Sep 11 8:49 PM 3

4 Sketch the graph of each function. 3. f(x) = 6 x 4. f(x) = 5 x 5. Jul 29 7:02 PM Oct 12 8:54 AM 4

5 Sep 7 8:11 PM Transformations use the same principals already learned. Sep 11 9:12 PM 5

6 Jul 29 7:15 PM Example 2: Sketch the graph using parent functions and the transformations Jul 29 7:09 PM 6

7 Natural base: e = most real world applications involving exponential functions use base e, instead of base 10 or 2. By evaluating the value of for greater and greater values of x, we can estimate that the value of this expression approaches a number close to , the irrational number called e. e = Sep 11 9:06 PM Transformations of y = e x follow the same rules. Jul 29 7:16 PM 7

8 Jul 29 7:37 PM Example 3: Graph. 13. y = 2e x 14. y = e 3x 15. y = e x y = e x 17. y = e x 5 Jul 29 7:18 PM 8

9 Jan 10 3:22 PM Exponential Growth and Decay: a common application of exponential growth is compound interest. Suppose an initial principal P is invested into an account with an annual interest rate r, and the interest is compounded or reinvested annually. At the end of each year, the interest earned is added to the account balance. This sum becomes the new principal for the next year. Aug 11 4:46 PM 9

10 Compound interest uses the limit formula for e with a little adjusting. A = the amount or the balance P = principal invested r = rate (change to a decimal) n = number of times compounded t = time (in years) Sep 12 6:12 AM Jul 29 7:38 PM 10

11 Example 4: 18. Mrs. Salisman invested $2000 into an educational account for her daughter when she was an infant. The account has a 5% interest rate. If Mrs. Salisman does not make any other deposits or withdrawals, what will the account balance be after 18 years if the interest is compounded: a. quarterly? b. monthly? c. daily? Jul 29 7:26 PM 19. If $1000 is invested in an online savings account earning 8% per year, how much will be in the account at the end of 10 years if there are no other deposits or withdrawals and interest is compounded: a. semiannually? b. quarterly? c. daily? Sep 11 8:32 PM 11

12 Notice as the number of compoundings increase, the balances also increase. But it is relatively small and it is less as the compoundings increase. Aug 11 4:52 PM Continuous compound interest. Jul 29 7:36 PM 12

investment. What will be her account balance after 18 years? 21.

13 Jul 29 7:36 PM Example 5: Continuous compounding: A = Pe rt 20. Mrs. Salisman found an account that will pay the 5% interest compounded continuously on her $2000 educational investment. What will be her account balance after 18 years? 21. If $1000 is invested in an online savings account earning 8% per year compounded continuously, how much will be in the account at the end of 10 years if there are no other deposits or withdrawals? Sep 12 6:30 AM 13

Continuous growth or decay is similar to continuous

14 Continuous growth or decay is similar to continuous compound interest. The growth or decay is compounded continuously rather than just yearly, monthly, hourly, or at some other time interval. Population growth can be modeled exponentially, continuously, and by other models. Sep 12 6:27 AM Jul 29 7:43 PM 14

Example 6: Sep 15 1:30 PM 22. A state's population is declining at a rate of 2.6% annually. The state currently has a population of approximately 11 million people.

15 Example 6: Sep 15 1:30 PM 22. A state's population is declining at a rate of 2.6% annually. The state currently has a population of approximately 11 million people. If the population continues to decline at this rate, predict the population of the state in 15 and 30 years. 23. The population of a town is declining at a rate of 6% if the current population is 12,426 people, predict the population in 5 and 10 years using each model. a. annually b. continuously Aug 11 5:04 PM 15

Answers are on next slide. Graphs follow.

Answers are on next slide. Graphs follow. Sec 3.1 Exponential Functions and Their Graphs November 27, 2018 Exponential Function - the independent variable is in the exponent. Model situations with constant percentage change exponential growth