1. Exponential Functions
Chapter 4
Exponential Functions

2. 4.1 Properties of Exponents
Know the meaning of exponent, zero exponent and negative exponent.
Know the properties of exponents.
Simplify expressions involving exponents
Know the meaning of exponential function.
Use scientific notation.

3. Exponent For any counting number n,
We refer to as the power, the nth power of b, or b raised to the nth power.
We call b the base and n the exponent.

4. Examples When taking a power of a negative number,
if the exponent is even the answer will be positive
if the exponent is odd the answer will be negative

5. Properties of Exponents
Product property of exponents
Quotient property of exponents
Raising a product to a power
Raising a quotient to a power
Raising a power to a power

6. Meaning of the Properties
Product property of exponents
Raising a quotient to a power

7. Simplifying Expressions with Exponents
An expression is simplified if:
It included no parenthesis
All similar bases are combined
All numerical expressions are calculated
All numerical fractions are simplified
All exponents are positive

8. Order of Operations Parenthesis Exponents Multiplication Division
Addition
Subtraction

9. Warning
Note: When using a calculator to equate powers of negative numbers always put the negative number in parenthesis.
Note: Always be careful with parenthesis

10. Examples

11. Examples (Cont.)

12. Zero Exponent
For b ≠ 0,
Examples,

13. Negative Exponent If b ≠ 0 and n is a counting number, then
To find , take its reciprocal and switch the sign of the exponent
Examples,

14. Negative Exponent (Denominator)
If b ≠ 0 and n is a counting number, then
To find , take its reciprocal and switch the sign of the exponent
Examples,

15. Simplifying Negative Exponents

16. Exponential Functions
An exponential function is a function whose equation can be put into the form:
Where a ≠ 0, b > 0, and b ≠ 1.
The constant b is called the base.

17. Exponential vs Linear Functions
x is a exponent
x is a base

18. Scientific Notation A number written in the form:
where k is an integer and
-10 < N ≤ -1 or
1 ≤ N < 10
Examples

19. Scientific to Standard Notation
When k is positive move the decimal to the right
When k is negative move the decimal to the left
move the decimal 3 places to the right
move the decimal 5 places to the left

20. Standard to Scientific Notation
if you move the decimal to the right, then k is positive
if you move the decimal to the left, then k is negative
move the decimal 4 places to the left
move the decimal 9 places to the right

21. Group Exploration
If time,
p173

22. 4.2 Rational Exponents

23. Rational Exponents ( ) For the counting number n, where n ≠ 1,
If n is odd, then is the number whose nth power is b, and we call the nth root of b
If n is even and b ≥ 0, then is the nonnegative number whose nth power is b, and we call the principal nth root of b.
If n is even and b < 0, then is not a real number.
may be represented as

24. Examples ½ power = square root ⅓ power = cube root
not a real number since the 4th power of any real number is non-negative

25. Rational Exponents
For the counting numbers m and n, where n ≠ 1 and b is any real number for which is a real number,
A power of the form or is said to have a rational exponent.

26. Examples

27. Properties of Rational Exponents
Product property of exponents
Quotient property of exponents
Raising a product to a power
Raising a quotient to a power
Raising a power to a power

28. Examples

29. 4.3 Graphing Exponential Functions

30. Graphing Exponential Functions by hand-3
1/8 -2
1/4 -1
1/2 1 2 4 3 8

31. Graph of an exponential function is called an exponential curve

32. x y -1 8 4 1 2 3
1/2

34. Base Multiplier Property
For an exponential function of the form
If the value of the independent variable increases by 1, then the value of the dependent variable is multiplied by b.

35. x increases by 1, y increases by b-3
1/8 -2
1/4 -1
1/2 1 2 4 3 8 x y -1 8 4 1 2 3
1/2

36. Increasing or Decreasing Property
Let , where a > 0.
If b > 1, then the function is increasing
grows exponentially
If 0 < b < 1, then the function is decreasing
decays exponentially

37. Intercepts
y-intercept for the form:
is (0,a)
is (0,1)

38. Intercepts Find the x and y intercepts: y-intercept x-intercept
as x increases by 1, y is multiplied by 1/3.
infinitely multiplying by 1/3 will never equal 0
as x increases, y approaches but never equals 0
no x-intercept exists, instead the x-axis is called the horizontal asymptote

39. Reflection Property The graphs
are reflections of each other across the x-axis
a > 0
a > 0
a < 0
a < 0

40. 4.4 Finding Equations of Exponential Functions

41. Finding an Equation Using a Table
Refer to the Base Multiplier Property
as x increases by 1, y is multiplied by the base
Find the y-intercept
(0, a) = (0, 2)
Find the constant
multiplying by 4
Write the equation x
F(x) 2 1 8 32 3
148 4

42. Linear vs Exponential x F(x) 243 1 81 2 27 3 9 4 x F(x) 81 1 70 2 59 3
243 1 81 2 27 3 9 4 x
F(x) 81 1 70 2 59 3 48 4 37

43. Solving for b
No Real Solutions

44. When Solving For b For , For ,
n is even, always have a positive and negative answer
n is odd, always have one positive answer
For ,
n is even, always no real solutions
n is odd, always have one negative answer

45. Solving for b (cont)
Remember to always simplify both sides of the equations first.

46. Finding an Equation of Exponential Curves Using Two Points
Given (0,4) and (5,128)
We know (0,4) is y-intercept (0,a) so a = 4
Substitute (5, 128) into the equation

47. Example (2,1) (5,7) Plug both points into the standard equation
Divide to cancel a term

48. Example (cont) Now we have b
We can substitute one of the points and solve for a …..(2,1)

49. 4.5 Using Exponential Functions to Model Data

50. Exponential Models
Exponential model – exponential function, or its graph, that describes the relationship between two quantities in an authentic situation
Exponentially related – If all the data points for a situation lie on an exponential curve
Approximately exponentially related – If all the data points lie on or close to an exponential curve

51. Suppose there are 5 million bacteria on a banana at 8am on Monday
Suppose there are 5 million bacteria on a banana at 8am on Monday. Every bacterium divides into 2 every hour, on average.
Give a function for the situation
Predict the number of bacteria at 8pm on Monday.
20,480 million bacteria

52. Exponential Functions of Time
y is the amount
t amount of time
a is the initial amount when t =0
b is constant by which a grows or decays over time

53. Write an equation to model the data
A person invests 8,000 in an account and interest is compounded 5% annually.
Write an equation to model the data
Value is 100% of the original deposit plus 6% after the first year so we use 1.06 as our base
What is the value after 10 years?
The value would be $14, after 10 years

54. Half-life
If a quantity decays exponentially, the half-life is the amount of time it takes for that quantity to be reduced to half
half life

55. Californium-251 is a radioactive element with a half-life of 900 years.
Give an equation to model the data.
What percent would be left after 600 years?
about 63%

56. Base of Exponential Model
b > 1, grows exponentially by a rate of
b – 1
0 < b < 1, decays exponentially by a rate of 1 – b
Notice: b ≠ 1, always equals one therefore the equation would be a horizontal line at the initial amount (a)

57. A person’s heart attack risk can be estimated by using Framingham point scores. If men’s risk of a heart attack is 1% at a Fram-score of 0, and 20% at a Fram-score of 15. Given they are exponentially linearly related, find an equation to model the data.
(0,1) and (15, 20)
a = 1
Predict the percent risk for a person with a score of 20

58. Exponential Functions from Tables
A person’s heart attack risk can be estimated by using Framingham point scores. Men’s risk of having a heart attack in the next 10 years is shown in the table.
Framingham Score
Risk)
(percent 1 5 2 10 6 15 20 17 30

59. Plot a Scattergram

60. Chose two points Divide the equations Find a (5,2) (15,20)
b represents the percent risk grows exponentially by 26% for each score point
a represents the approximate initial percent risk at a score of 0 points

61. Graph line over scattergram to check