E XPONENTIAL F UNCTIONS GET A GUIDED NOTES SHEET FROM THE FRONT!

E XPONENTIAL F UNCTIONS An exponential function is a function with a variable in the exponent. f(x) = a(b) x

E XPONENTIAL F UNCTIONS 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

L ETS 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.

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

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

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.

I DENTIFY 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

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

A SYMPTOTES 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.

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

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

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

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

E XPONENTIAL G ROWTH 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.

G ROWTH G RAPHS Of the following, which graphs show exponential GROWTH?

E VALUATING AN E XPONENTIAL F UNCTION 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 w eeks. How many beetles will there be after 56 days? Step 1: Convert 56 days to weeks. Step 2: Evaluate for x = 8.

E XAMPLE : 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?

Y OU T RY 3. An initial population of 20 rabbits triples every half year. The function gives the population after x half-year periods. How many rabbits will there be after 3 years?

E XPONENTIAL D ECAY 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.

G ROWTH G RAPHS Of the following, which graphs show exponential DECAY?

E XPONENTIAL G ROWTH /D ECAY 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.

E XAMPLES 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?

E XAMPLES 2. The kilopascal is a unit of measure for atmospheric pressure. The atmospheric pressure at sea level is about 101 kilopascals. For every 1000-m increase in altitude, the pressure decreases about 11.5%. What is the approximate pressure at an altitude of 3000 m?

Y OU T RY 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.

Y OU T RY 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.

H OMEWORK Worksheet

W ARM U P M ARCH 20 TH 1. Is the following equation exponential growth or decay? y = 3/2 · 3 x

B ASIC & C OMPOUND I NTEREST

E XPONENTIAL G ROWTH /D ECAY 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 change 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.

E XAMPLES 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?

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.

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.

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

M&M S Move your desks into groups of 4. Make sure all group members’ names are on the worksheet. You only have until the end of class. Do not eat the M&Ms until you are DONE!

H OMEWORK Worksheet