EXPONENTIAL FUNCTIONS GET A GUIDED NOTES SHEET FROM THE BACK!

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1 EXPONENTIAL FUNCTIONS GET A GUIDED NOTES SHEET FROM THE BACK!

2 EXPONENTIAL FUNCTIONS An exponential function is a function with a variable in the exponent. f(x) = a(b) x

3 EXPONENTIAL FUNCTIONS Parent graphs of exponential functions are in the form: f(x) = b x Parent function- original function before any changes have been made. for example: f(x) = 4 x

4 LETS REVIEW THE SHIFTS THAT OCCUR WITH IN EXPONENTIAL FUNCTIONS. Original Function: f(x) = b x f(x) = -b x Negatives in front cause a reflection across the x-axis.

5 Original Function: f(x) = b x f(x) = b x-1 f(x) = b x+1 right left Numbers in the exponents cause horizontal shifts (subtract right, or add left).

6 Original Function: y = b x f(x) = b x - 1 f(x) = b x + 1 down up Numbers behind the original function cause vertical shifts (subtrct down, and add up).

7 Original Function: f(x) = b x f(x) = a(b) x Numbers larger than 1 that are in front of the b value cause a stretch.

8 IDENTIFY THE PARENT FUNCTION OF EACH, AND THE TRANSFORMATIONS: 1. f(x) = 3 x 8 2. f(x) = -3(2) x 3. f(x) = 4 x+5 4. f(x) = 2 x f(x) = 5 x-2 PARENT TRANSFORMATIONS

9 F(X) = 3(2) X Parent Graph: Transformation: Domain: Range: X Y

10 ASYMPTOTES All exponential functions have horizontal asymptotes. Notice that the range values of the previous graph were restricted by the horizontal asymptote. Range is always restricted by the asymptotes.

11 F(X) = 4 X-3 X Parent Graph: Transformation: Y Domain: Range:

12 F(X) = -8(.5) X Parent Graph: Transformation: Domain: Range: X Y

13 F(X) = 3 X + 2 Parent Graph: Transformation: Domain: Range: X Y

14 F(X) = -2 X - 3 Parent Graph: Transformation: Domain: Range: X Y

15 EXPONENTIAL GROWTH Exponential growth is an initial amount that increases at a steady rate over time. Exponential growth can be modeled by the function , where a > 0 and b > 1. The base b is the growth factor, which equals 1 plus the percent rate of change expressed as a decimal.

16 GROWTH GRAPHS Of the following, which graphs show exponential GROWTH?

17 EXPONENTIAL DECAY Exponential Decay occurs when an initial amount decreases at a steady rate over time. Exponential decay can be modeled by the function , where a > 0 and b < 1. The base b is the decay factor, which equals 1 minus the percent rate of change expressed as a decimal.

18 GROWTH GRAPHS Of the following, which graphs show exponential DECAY?

19 EXPONENTIAL GROWTH/DECAY y = a (b) x Equation: A = P(1 ± r) t. A represents the final amount. P represents the initial amount. r represents the rate of growth/decay expressed as a decimal. t represents time. Key words to look for that tell you to use the formula is increase, appreciate and growth. Key words to look for that tell you to use the formula is decrease, depreciate and decay.

20 EVALUATING AN EXPONENTIAL FUNCTION 1. Example: Suppose 30 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles each week. The function gives the population after weeks. How many beetles will there be after 56 days? Step 1: Convert 56 days to weeks. Step 2: Evaluate for x = 8.

21 EXAMPLE: The amount of money spent at West Outlet Mall in Midtown continues to increase. The total T(x) in millions of dollars can be estimated by the function T(x)=12(1.12) x, where x is the number of years after it opened in a) According to the function, find the amount of sales in 2006, 2008 and b) Name the y-intercept. c) What does it represent in this problem?

22 EXAMPLES 1. The original price of a tractor was $45,000. The value of the tractor decreases at a steady rate of 12% per year. a. Write an equation to represent the value of the tractor since it was purchased. b. What is the value of the tractor in 5 years?

23 YOU TRY 3. Find a value of a $20,000 car in five years if it depreciates at a rate of 12% annually. Write the exponential function to model the situation, and find the amount after the specified time.

24 EXAMPLES 4. A population of 1,860,000 decreases 1.5% per year for 12 years. Write the exponential function to model the situation, and find the amount after the specified time.

25 EXAMPLES 1. POPULATION The population of Johnson City in 1995 was 25,000. Since then, the population has grown at an average rate of 3.2% each year. a. Write an equation to represent the population of Johnson City since b. According to the equation, what will the population of Johnson City be in the year 2005?

26 2. Find the current value of a $125,000 home that was purchased, in 2010, if it appreciates at a 4% rate annually. Write the exponential function to model the situation, and find the amount after the specified time.

27 BASIC & COMPOUND INTEREST

28 A r = P(1 ± ) n n t Daily = P =

29 3. The Lieberman s have $12,000 in a savings account. The bank pays 3.5% interest on savings accounts, compounded monthly. Find the balance in 3 years.

30 4. Determine the amount of an investment if $300 is invested, at an interest rate of 6.75%, compounded semiannually for 20 years.

31 COMPOUNDING CONTINUOUSLY A = Pe rt A = final amount P = initial amount R = rate T = time E = a button on your calculator (like pi) Key word: continuously

32 Suppose $5000 is put into an account that pays 4% compounded continuously. How much will be in the account after 3 years?

34 HOMEWORK Worksheet (Skip 7 & 9) Midterm Review Page

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