1. Objective
Students will be able to graph exponential growth functions.

2. Exponential Functions
An exponential function has the form y = abx where a ≠ 0 and the base b is a positive number other than 1.
What do you think makes a function an exponential growth function?
Think about the equations from the M&M’s activity.

3. If a > 0 and b > 1, then the function y = abx is an exponential growth function, and b is called the growth factor.
Parent Function for Exponential Growth Functions
The function f(x) = bx, where b > 1, is the parent function for the family of exponential growth functions with base b.
The domain of f(x) = bx is (-∞, ∞), the range is (0, ∞), and the asymptote is the line y = 0.

4. Graph y = 2x. Step 1: Make a table of values.
Step 2: Plot the points from the table.
Step 3: Draw, from left to right, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the right.
Domain: (-∞, ∞)
Range: (0, ∞)
Asymptote: y = 0

5. Graph the following functions and state domain, range, and asymptote.2) 3)
Domain: (-∞, ∞)
Range: (0, ∞)
Asymptote: y = 0
Domain: (-∞, ∞)
Range: (-∞, 0)
Asymptote: y = 0

6. Exponential Functions
An exponential function has the form y = abx where a ≠ 0 and the base b is a positive number other than 1.
b is the growth or decay
a is the initial value (y-intercept)

7. Translations
To graph a function of the form y = abx – h + k, begin by sketching the graph of y = abx. Then translate the graph horizontally by h units and vertically by k units.
Graph State the domain, range, and asymptote.
The graph’s asymptote is the line y = -3.
So domain: (-∞, ∞) and range (-3, ∞)
**asymptote is the line y = k

8. 5) Graph . Then state the domain, range, and asymptote.
Make a table of values and sketch the graph. x -3 -2 -1 1 y
19/9
7/3 3 5 11
Domain:
Range:
(-∞, ∞)
Asymptote:
y = 2
(2, ∞)

9. Homework
p. 482: 3, 4, 5, 7, 15, 17
**Find the asymptote as well

10. Objective
Students will be able to understand exponential growth in real life problems and compound interest.
Exponential Functions ( ) Quiz on Friday!

11. Exponential Growth Models
When a real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:
where a is the initial amount and r is the percent increase expressed as a decimal. Note that the quantity 1 + r is the growth factor.

12. In 1996, there were 2573 computer viruses and other computer security incidents. During the next 7 years, the number of incidents increased by about 92% each year.
Write an exponential growth model giving the number n of incidents t years after 1996.
About how many incidents were there in 2003?
Graph the model.
Use the graph to estimate
the year when there were
about 125,000 computer
security incidents.
a = 2573
r = 0.92
t = 7
computer security incidents
(t ≈ 6), so 2002

13. In 1977, there were 41 breeding pairs of bald eagles in Maryland
In 1977, there were 41 breeding pairs of bald eagles in Maryland. Over the next 24 years, the number of breeding pairs increased by about 8.9% each year.
Write a model giving the number n of breeding pairs of bald eagles in Maryland t years after 1977.
Make a table of values for the model.
Graph the model.
Use the graph to estimate
how many breeding pairs of bald
eagles were in Maryland in 1992.
a = 41
r = 0.089 t 6 12 18 n 41 68
114
190
(t = 15), so about 150 pairs of bald eagles

14. In the exponential growth model y = 527(1
In the exponential growth model y = 527(1.39)x, identify the initial amount, the growth factor, and the percent increase.
initial amount (a) = 527
growth factor (1 + r) = 1.39
percent increase (r%) = 39%

15. Compound Interest
Exponential growth functions are used in real-life situations involving compound interest. Compound interest is paid on the initial investment, called the principal, and on previously earned interest. Interest paid only on the principal is called simple interest.
Consider an initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal), compounded n times per year. The amount A in the account after t years is given by this equation:

16. You deposited $4000 in an account that pays 2. 92% annual interest
You deposited $4000 in an account that pays 2.92% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency.
Quarterly
Daily
The balance at the end of 1 year is $
P = 4000, r = ,
n = 4, t = 1
The balance at the end of 1 year is $
P = 4000, r = ,
n = 365, t = 1

17. You deposited $2000 in an account that pays 4% annual interest
You deposited $2000 in an account that pays 4% annual interest. Find the balance after 3 year if the interest is compounded daily.
The balance at the end of 3 years is $
P = 2000, r = 0.04,
n = 365, t = 3

18. Homework
p. 482: 25, 28, 29, 30, 35

19. Objective
Students will be able to graph and model exponential decay functions.
Exponential Functions ( ) Quiz on Friday!

20. If a > 0 and 0 < b < 1, then the function
y = abx is an exponential decay function, and b is called the decay factor.
Parent Function for Exponential Decay Functions
The function f(x) = bx, where 0 < b < 1, is the parent function for the family of exponential decay functions with base b.
The domain of f(x) = bx is (-∞, ∞), the range is (0, ∞), and the asymptote is the line y = 0.

21. Graph . Step 1: Make a table of values.
Step 2: Plot the points from the table.
Step 3: Draw, from right to left, a smooth curve that begins just above the x-axis, passes through the plotted points, and moves up to the left.
Domain: (-∞, ∞)
Range: (0, ∞)
Asymptote: y = 0

22. Graph the following functions and state domain, range, and asymptote.2) 3)
Domain: (-∞, ∞)
Range: (0, ∞)
Asymptote: y = 0
Domain: (-∞, ∞)
Range: (-∞, 0)
Asymptote: y = 0

23. Graph . State the domain, range, and asymptote. **Translation
Make a table of values and sketch the graph. x -2 -1 1 2 y 4
-1/2
-5/4
-13/8
Domain:
Range:
Asymptote:
(-∞, ∞)
(-2, ∞)
y = -2

24. Exponential Decay Models
When a real-life quantity decreases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by the equation:
where a is the initial amount and r is the percent decrease expressed as a decimal. Note that the quantity 1 – r is the decay factor.

25. A new snowmobile costs $4200
A new snowmobile costs $ The value of the snowmobile decreases by 10% each year.
Write an exponential decay model giving the snowmobile’s value y (in dollars) after t years.
b) What is the value after 3 years?
c) Graph the model.
d) Use the graph to estimate
when the value of the
snowmobile will be $2500.
a = 4200
r = 0.10
t = 3 $
t is between 4 and 5, so during the 4th year

26. The value of a car can be modeled by the equation
y = 24,000(0.845)t where t is the number of years since the car was purchased.
a) Graph the model.
b) Estimate when the value of the car will be $10,000.
c) Use the model to predict the value of the car after 50 years. Is this a reasonable value? Explain.
t is between 4 and 5, so during the 4th year
$5.29
No; because a car does not normally last 50 years

27. Homework p. 489: 3-6, 14, 19, 30, 31 Problem 30 is on next slide!
**Find the asymptote as well
Problem 30 is on next slide!

29. Warm Up
You invest $500 in the stock of a company. The value of the stock decreases by 2% each year. Describe and correct the error in writing a model for the value of the stock after t years.
The decay factor 1 – r, not r
y = 500(0.98)t

30. Homework Questions? 3) exponential decay 4) exponential growth
5) exponential growth 6) exponential decay
14) )
30) a) about mg 31) a) y = 200(0.75)t; about $84.38
b) about mg b)
c) about mg
c) after about
2.5 years
Domain: (-∞, ∞)
Range: (-1, ∞)
Asymptote: y = -1

31. Objective
Students will be able to understand continuously compounded interest.
Students will be able to correct their mistakes from their rational functions test.
Exponential Functions ( ) Quiz on Friday!

32. Continuously Compounded Interest
When interest is compounded continuously, the amount A in an account after t years is given by the formula:
where P is the principal and r is the annual interest rate expressed as a decimal.

33. The balance at the end of 1 year is $4247.35.
You deposit $4000 in an account that pays 6% annual interest compounded continuously. What is the balance after 1 year?
Write formula
Substitute 4000 for P, 0.06 for r, and 1 for t
Use a calculator
The balance at the end of 1 year is $

34. You deposit $2500 in an account that pays 5% annual interest compounded continuously. Find the balance after each amount of time.
2 years
5 years
7.5 years $ $ $

35. Homework
p. 497: 57, 58

36. Objective
Students will be able to understand the natural base e functions.

37. Natural Base e
The natural base e is irrational. It is defined as follows:
As n approaches +∞,
approaches e ≈

38. Simplify the following expressions.1) 2) 3)

39. Simplify the following expressions.4) 5) 6) 7)

40. Natural Base Functions
A function of the form y = aerx is called a natural base exponential function.
If a > 0 and r > 0, the function is an exponential growth function
If a > 0 and r < 0, the function is an exponential decay function
The graphs of the basic functions y = ex and y = e-x are shown below.

41. where t is the shark’s age (in years).
The length l (in centimeters) of a tiger shark can be modeled by the function
where t is the shark’s age (in years).
What is the length of a tiger shark that is 3 years old?
What is the length of a tiger shark that is 5 years old?
How old is a tiger shark
that is 281 centimeters long?
about 175 centimeters
about 224 centimeters
about 9 years old

42. The percent L of surface light that filters down through bodies of water can be modeled by the exponential function where k is a measure of murkiness of water and x is the depth below the surface (in meters).
A recreational submarine is traveling in clear water with a k-value of about Write an equation giving the percent of surface light that filters down through clear
water as a function of depth.
b) What is the percent of surface light available at
a depth of 40 meters?
c) How deep can the submarine descend in clear
water before only 50% of surface light is
available?
about 45%
about 35 meters

43. Homework
p. 495: 5, 12, 13, 14, 55, 56

44. Objective
Students will be able to evaluate logarithms.

45. Solve for x in the following problems.1) 2) 3)
x = 3
x = 4
x = -2

46. Logarithm with Base b
Let b and y be positive numbers with b ≠ 1. The logarithm of y with base b is denoted by
and is defined as follows:
if and only if
The expression is read as “log base b of y.”

47. Logarithm with Base b
is in exponential form where b is the base, x is the exponent, and y is the solution
is in logarithmic form where b is the base, x is the solution, and y is the number you are taking the log of

48. Convert from exponential to logarithmic form:
1) 32 = 9
b = 3 x = 2 y = 9
2) 24 = 16
b = 2 x = 4 y = 16 3)
b = ¼ x = -3 y = 64

49. Convert from logarithmic form to exponential form:
Swirl method: 1)
b = 9 x = 2 y = 81
92 = 81 2) 3)
121 = 12 4)
40 = 1

50. Let b be a positive real number such that b ≠ 1.
Logarithm of 1:
Logarithm of b with Base b:
because
**Number 4 from previous slide
because
**Number 3 from previous slide

51. Evaluate the following logarithms.
**Ask yourself what power of b gives you y?
4 to what power gives 64?
4x = 64, what is x? 1) 2) 3) 4)
43 = 64, so log464 = 3
5 to what power gives 1/5?
5x = 1/5, what is x?
5-1 = 1/5, so log5(1/5) = -1 -3
1/5x = 125, what is x?
1/2

52. Special Logarithms
A common logarithm is a logarithm with base 10. It is denoted by log10 or simply by log. A natural logarithm is a logarithm with base e. It can be denoted by loge, but is more often denoted by ln.
Common Logarithm:
Natural Logarithm:
Use a calculator to evaluate the logarithm:
1) log 8
2) ln 0.3
≈ 8
e ≈ 0.3

53. Homework
p. 503: 3-15 (odds), 21, 23

54. Objective Students will be able to find inverse functions.
Homework Quizzes; grades will be entered into your homework category; they are short (5 minute) quizzes; you will be warned ahead of time when you have one

55. Inverse Functions
By the definition of a logarithm, it follows that the logarithmic function
is the inverse of the exponential function
This means that ✔
and ✔

56. Use inverse properties to simplify the following expressions:1) 2) 3)
Express 25 as a power with base 5
Power of a power property

57. Find the inverse of the following functions:
Switch x and y 1) 2)
Write in logarithmic form (solve for y)
The inverse of is
Switch x and y
Write in exponential form (solve for y)
The inverse of is

58. Inverse Recall: When taking an inverse, you first flip x and y.
If you start with a function in exponential form,
what form should the inverse be in?
If you start with a function in logarithmic form,
**Switch your domain and range
logarithmic form
exponential form
logarithms and exponents are inverses of one another; they undo each other

59. Simplify the expression.
1) )
3) )
5) Find the inverse of
6) Find the inverse of x
-3x 6x 20
y = log4x
y = ex

60. Homework
p. 504: (odds), 37 – 39