1. Intro to Exponential Functions
Lesson 3.1
F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

2. Monday, March 9, 2015

3. (TRY IT without using a calculator)
Monday Bellwork
Solve for ‘y’
(TRY IT without using a calculator)
1. y = 30 =
2. y = 2-1 =
3. y = 23 =
4. y = 4-2 =
5. 1 = 2y

4. Linear vs Exponential Functions Contrast
Linear Functions
Change at a constant rate
Rate of change (slope) is a constant
Exponential Functions
Change at a changing rate
Change at a constant percent rate

5. Contrast Suppose you have a choice of two different jobs at graduation
Start at $30,000 with a 6% per year increase
Start at $40,000 with $1200 per year raise
Which should you choose?
One is linear growth
One is exponential growth

6. Which Job? How do we get each next value for Option A?
Year
Option A
Option B 1
$30,000
$40,000 2
$31,800
$41,200 3
$33,708
$42,400 4
$35,730
$43,600 5
$37,874
$44,800 6
$40,147
$46,000 7
$42,556
$47,200 8
$45,109
$48,400 9
$47,815
$49,600 10
$50,684
$50,800 11
$53,725
$52,000 12
$56,949
$53,200 13
$60,366
$54,400 14
$63,988
$55,600
How do we get each next value for Option A?
When is Option A better?
When is Option B better?
Rate of increase a constant $1200
Rate of increase changing
Percent of increase is a constant
Ratio of successive years is 1.06

7. Amount of interest earned
Example
Consider a savings account with compounded yearly income
You have $100 in the account
You receive 5% annual interest
At end of year
Amount of interest earned
New balance in account 1
100 * 0.05 = $5.00
$105.00 2
105 * 0.05 = $5.25
$110.25 3
* 0.05 = $5.51
$115.76 4 5
View completed table

8. Compounded Interest
Completed table

9. Compounded Interest Table of results from calculator Graph of results
Set y= screen y1(x)=100*1.05^x
Choose Table (Diamond Y)
Graph of results

10. Exponential Modeling
Population growth often modeled by exponential function
Half life of radioactive materials modeled by exponential function

11. Growth Factor
Recall formula new balance = old balance * old balance
Another way of writing the formula new balance = 1.05 * old balance
Why equivalent?
Growth factor: interest rate as a fraction

12. Decreasing Exponentials
Consider a medication
Patient takes 100 mg
Once it is taken, body filters medication out over period of time
Suppose it removes 15% of what is present in the blood stream every hour
At end of hour
Amount remaining 1
100 – 0.15 * 100 = 85 2
85 – 0.15 * 85 = 72.25 3 4 5
Fill in the rest of the table
What is the growth factor?

13. Decreasing Exponentials
Completed chart
Graph
Growth Factor = 0.85
Note: when growth factor < 1, exponential is a decreasing function

14. Solving Exponential Equations Graphically
For our medication example when does the amount of medication amount to less than 5 mg
Graph the function for 0 < t < 25
Use the graph to determine when

15. General Formula All exponential functions have the general format:
Where
A = initial value
B = growth factor
t = number of time periods

16. Typical Exponential Graphs
When B > 1
When B < 1
View results of B>1, B<1 with spreadsheet

17. I will graph a basic function
f(x) = 2x
Pre Calc Cookbook:
1. Make a table of at least 5 (x,y) values
2. Determine intercept(s)
3. Draw it on a graph!!

18. WE graph a basic function
f(x) = 2-x
Pre Calc Cookbook:
1. Make a table of at least 5 (x,y) values
2. Determine intercept(s)
3. Draw it on a graph!!

19. YOU graph a basic function
f(x) = 2x + 3
Pre Calc Cookbook:
1. Make a table of at least 5 (x,y) values
2. Determine intercept(s)
3. Draw it on a graph!!

20. Exponential Functions HW 1

21. Tuesday, March 10, 2015

22. Tuesday Bellwork
Think pair share the HW from last night! THEN: Graph the function y = 3-x

23. Definition of the Exponential Function
The exponential function f with base b is defined by
f (x) = bx or y = bx
Where b is a positive constant, then x is any real number.
Here are some examples of exponential functions.
f (x) = 2x g(x) = 10x h(x) = 3x+1
Base is 2.
Base is 10.
Base is 3.

24. Characteristics of Exponential Functions
The domain of f (x) = bx consists of all real numbers. The range of f (x) = bx consists of all positive real numbers.
The graphs of all exponential functions pass through the point (0, 1) because f (0) = b0 = 1.
If b > 1, f (x) = bx has a graph that goes up to the right and is an increasing function.
If 0 < b < 1, f (x) = bx has a graph that goes down to the right and is a decreasing function.
f (x) = bx is a one-to-one function and has an inverse that is a function.
The graph of f (x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote.
f (x) = bx
0 < b < 1
f (x) = bx
b > 1

25. Transformations Involving Exponential Functions
Shifts the graph of f (x) = bx upward c units if c > 0.
Shifts the graph of f (x) = bx downward c units if c < 0.
g(x) = -bx + c
Vertical translation
Reflects the graph of f (x) = bx about the x-axis.
Reflects the graph of f (x) = bx about the y-axis.
g(x) = -bx
g(x) = b-x
Reflecting
Multiplying y-coordintates of f (x) = bx by c,
Stretches the graph of f (x) = bx if c > 1.
Shrinks the graph of f (x) = bx if 0 < c < 1.
g(x) = c bx
Vertical stretching or shrinking
Shifts the graph of f (x) = bx to the left c units if c > 0.
Shifts the graph of f (x) = bx to the right c units if c < 0.
g(x) = bx+c
Horizontal translation
Description
Equation
Transformation

26. Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.
Example 1:
Use the graph of f (x) = 3x to obtain the graph of g(x) = 3 x+1.
Solution Examine the table below. Note that the function g(x) = 3x+1 has the general form g(x) = bx+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs.
f (x) = 3x
g(x) = 3x+1
(0, 1)
(-1, 1) 1 2 3 4 5 6 -5 -4 -3 -2 -1

27. Note: This is similar to section I on HW.
Problems
Sketch a graph using transformation of the following: 1. 2. 3.
Recall the order of shifting: horizontal, reflection (horz., vert.), vertical.
Note: This is similar to section I on HW.

28. The domain of f (x) = bx consists of all real numbers.
Sketch the graph of the function WITHOUT the graphing calculator. Then give the domain and range in interval notations for each graph. Also give the intercept(s).
The domain of f (x) = bx consists of all real numbers.
The range of f (x) = bx consists of all positive real numbers.
Y-intercept, let x=0 and solve for y!
X-intercept, let y=0 and solve for x! (does not always exist)

29. Example 1: Domain: all real numbers Range: all positive real numbers
Note: This is similar to section II on HW.
Sketch the graph of the function WITHOUT the graphing calculator. Then give the domain and range in interval notations for each graph. Also give the intercept(s).
Domain: all real numbers
Range: all positive real numbers
Intercept:

30. EX 2: Determining the transformation between two functions!
f(x)= 2x and g(x)= 2x-3
Shifts the graph of f (x) = 2x to the right 3 units because -3 < 0
Note: This is similar to section III on HW.

31. EX 3: Determining the transformation between two functions!
f(x)= - 3x and g(x)= 6 - 3x
Shifts the graph of f (x) = -3x upward 6 units because 6 > 0.
Note: This is similar to section III on HW.

32. Independent Practice
Complete sections 1-3

33. Wednesday, March 11, 2015

34. Bellwork Pair-Share Sections 1-3 on your homework form last night
We will review this briefly

35. Note: This is similar to section IV on HW.
One-To-One Property
For a>0 and a ≠ 1,
if am = an , then m = n
Note: This is similar to section IV on HW.

36. Example 1: Pre Calc Cookbook: Change fractions to numerator only
Find common bases (if not already)
Set exponents equal to each other
Solve for your missing variable

37. Example 2: Pre Calc Cookbook: Change fractions to numerator only
Find common bases (if not already)
Set exponents equal to each other
Solve for your missing variable

38. Example 3: Pre Calc Cookbook: Change fractions to numerator only
Find common bases (if not already)
Set exponents equal to each other
Solve for your missing variable

39. The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately,
The number e is called the natural base. The function f (x) = ex is called the natural exponential function.
f (x) = ex
f (x) = 2x
f (x) = 3x
(0, 1)
(1, 2) 1 2 3 4
(1, e)
(1, 3) -1

40. Formulas for Compound Interest
After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas:
For n compoundings per year:
For continuous compounding: A = Pert.

41. Example: Choosing Between Investments
You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?
Solution We use the compound interest model with P = 8000,
r = 7% = 0.07, n = 12 (monthly compounding, means 12 compoundings per year), and t = 6.
The balance in this account after 6 years is $12,
Note: This is similar to section V on HW.
more

42. Example: Choosing Between Investments
You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment?
Solution For the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = , and t = 6.
The balance in this account after 6 years is $12,066.60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option.

43. Example
Use A= Pert to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously
Solution:
A= Pert A = 2000e(.07)(8) A = 2000 e(.56) A = 2000 * 1.75 A = $3500

44. Independent Practice
Complete sections 4 & 5