1. Put post-writes in the box.
Warm-up (pg. 45, 7 minutes)
Put post-writes in the box.
A rich man (not a teacher!) offers you a choice of prizes to reward
your hard work in Algebra 2. You can either have:
a) $1,000/day for a month (30 days)
b) 1 cent on the first day that will double each day for 30 days
Which would you choose? Why?

2. Exponential Functions Discovery (pg. 45)
Evaluate and graph the following functions using your calculator:
1) For f(x) = 3
a. Find f(0) b. Find f(1) c. Find f(2) d. Find f(3)
2) For f(x) = (0.8)
3) What do you notice in Problem #1 as x increases?
4) What do you notice in Problem #2 as x increases?
5) Can we draw any conclusions about the relationships between the equations values and
the graphs? x x

3. Exponential Functions
y = ab
x - Independent Value (time)
y - Dependent Value (final amount)
a - Principal (starting amount, y-intercept)
b - Growth or Decay Factor
If b > 1, the equation is growing. If a rate of
change (r) is given for an increase, b = 1 + r
(where r is a decimal - NOT a percent!)
If b < 1, the equation is decreasing. If a rate of
decay (r) is given for a decrease, b = 1 - r (where
r is a decimal - NOT a percent!)

4. Exponential Growth
For the first 12 months, a baby's height grows exponentially by 7% each
month. If a baby was born 23 inches tall, how tall is it after 8 months?
After how many months will the baby be 30 inches tall?
How long after the baby is born will its height double?

5. Exponential Decay
With our new fiscally conservative government, the national debt is
falling by 8% each year. If the debt started out at $11 trillion, how big
will it be after 10 years?
If the debt keeps falling at the same rate, when will it be under $2 trillion?
When will the debt level be at ¼ of the current level?

6. Consider the function: f(x) = ab .
Argumentation x
Consider the function: f(x) = ab .
Based on what we learned in Unit 1 about y-intercepts,
why is a the y-intercept of the equation?
(Standard Hint: substitute values in for a, b, and x to see
the relationship!)

7. Warm-up

8. Graphing Exponential Functions Discovery:
1) Graph y = 2 on your calculator. x x
2) Graph y = 5 on your calculator. x
3) Graph y = 10 on your calculator.
4) What is the y-intercept of all these graphs? Why?
5) What is the x-intercept of all these graphs? Why?

9. Graphing Exponential Functions Discovery:
1) Graph y = 2 on your calculator. x
x + 4
2) Graph y = on your calculator.
(Remember to put x + 4 in parentheses.) x
3) Graph y = on your calculator.
(Do not put the +4 in parentheses.)
4) How is the second graph different from the first?
5) How is the third graph different from the first?

10. The amount of time it takes HALF of a material to decay.
Half-Life
Half-Life
The amount of time it takes
HALF of a material to decay. t
14.3
The half-life of phosphorus-32,
used in plant fertilizer, is 14.3
days. Write a function to model
the decay of a 50 mg sample.
After 84 days, how much of the
sample is left?
When will only 5 mg remain?
f(x) = a (½)

11. Doubling Time
Doubling Time
The amount of time it takes
a value to DOUBLE its growth.
At a certain interest rate,
an account's value doubles
in 8 years. Write a function
to model a $20,000 initial
investment.
What will be the value
after 30 years?
When will the account be
worth $200,000?

12. Hurricanes
How long will it take until everyone has their power back?
Complete this activity to figure it out!
Hurricane Activity

13. Warm-up

14. Compound Interest
Compounding Interest - Interest that is calculated more often than once per year
Jane deposits $ 200 into her account that is compounded quarterly with
8% annual interest. If she does not deposit any additional funds, how
much money will Jane have in 5 years?
How many times is interest calculated over 5 years?
What is the percentage rate for each interest calculation?
Interest Compounded Annually
Formula
Interest Compounded Quarterly
Formula

15. Compounds in a length of time:
Compounding Interest
Compounds in a length of time: nt r n
A = P(1 + )
A = P = r =
n = t =

16. In calculus, a limit is the number that a function approaches
e Discovery
In calculus, a limit is the number that a function approaches
as the value of the variable approaches a number or infinity.
Today, we are going to calculate the limit as n approaches
infinity of:
f(x) = ( ) n 1 n
Evaluate, to 5 decimal places:
f(1) = f(2) = f(3) = f(4) =
f(10) = f(100) = f(1000) = f(10,000) =

17. e Value
The limit as n got bigger and bigger approaches
This is called the natural logarithm, or e.
e is a NUMBER. What other letters represent numbers?

18. Compounding Continuously
Compounds CONTINUOUSLY: rt
A = Pe
A = P = e =
r = t =
A woman invests $200 at an 8% interest rate, compounded continuously.
How much money does she have after 5 years? How does this compare to
when the interested was compounded annually and quarterly?

19. Argumentation
1. If you invest $10,000 from the time you are born until you go to
college (18 years) at a 6% interest rate, how much money will you
have if the interest:
a) Compounds annually?
b) Compounds monthly?
c) Compounds daily?
d) Compounds continuously?
2. In your own words, why do the slight changes affect the answer so
much when growth is exponential?

20. Complete the chart for f(x) = 2 . What do you notice about the values?
Warm-up x
Complete the chart for f(x) = 2 . What do you notice about
the values?

21. Logarithms (Logs and Lns)
A logarithm is an inverse of an exponential function. x
Log a = x
Log a = x
Ln a = x
means
b = a
10 = a
e = a b x x
1) Rewrite log 32 in exponential form and solve.
2) Solve log 1000. x
3) Rewrite e = 27 in logarithmic form. 2 x
4) Write 8 = 19 in logarithmic form.

22. Evaluating Logs - Tic Tac Toe
Evaluate the following logs - but no calculators!

23. Complete the chart for g(x) = log x . Explain any relationships
Argumentation
Complete the chart for g(x) = log x . Explain any relationships
between the domain, range, and characteristics. 2
g(x)

24. Warm-up

25. Unfortunately, our calculators can only compute logs in base 10 or
Change of Base Formula
Unfortunately, our calculators can only compute logs in base 10 or
base e. We have to have a way to "change the base."
log x
log b
log x = k b k
The "k" we will use will be either ____ or ____.
1) log ) log ) log ) log 3 6 3
1.04

26. Solving Exponential Equations
1) 8 = 120
Graphing
Change of Base

27. Solving Exponential Equations
1) 3(8) = 450
2) 1000 (1.05) = 2000

28. Problem Task

29. 1) 4 - 5 = 17 2) If the population of 3) If you invest $25,000
Warm-up (10 min, pg. 51)
Solve the following:
1) = ) If the population of 3) If you invest $25,000
Charlotte is 3 million, at a 4% interest rate,
how long will it take how long will it take the
the population to money to double?
double?
(The population is
growing at a rate of
7.3% per year.) x

30. Paideia Seminars 1. Guidelines: 2. Goal setting:
- Don't raise hands, but be respectful with speech
- Refer to the text and examples
- Refer to other participants' comments
- Apply group norms to whole class discussion
2. Goal setting:
- Personal goal (Write it down!)
- Class goal

31. Text For Paideia
We have learned several properties for evaluating logarithms and
solving exponential equations. On pg. 51, where we solved the
warm-up, please write down the change of base formula.
The text we will use for this Paideia will be the change of base
formula and the process for solving problems #2 and 3 of the
warm-up.

32. Paideia Questions to Answer
1. Based on Problem #1 and the Change of Base formula, what is a
shortcut for solving exponential equations?
2. Based on Problems #2 and 3, what is a shortcut for finding out
how long it will take a value to double, triple, etc.?

33. Solve the following problems using the shortcut from Question #1.
Post-Write
Solve the following problems using the shortcut from Question #1. x
1) 8 = ) If you invest $5,000 at a 6% interest
rate, how long will it take until you have
$12,000?
Did the shortcut make the problems easier for you? Will you use it
in the future? Why or why not?
Solve the following problems using the shortcut from Question #2.
1) If you invest $8,000 at 2) A car's is valued at $18,000
a 3% interest rate, how long when it is new. If it depreciates at a
will it take the money to rate of 18% per year, how many
double? years will it take until it is worth
50% of its initial value?
Did the shortcut make the problems easier for you? Will you use it
in the future? Why or why not?

34. Warm-up

35. Product Property: log xy = log x + log y
Properties of Logs
Product Property: log xy = log x + log y
Quotient Property: log = log x - log y
Power Property: log x = a log x x y a 2 x 3y
1) Expand: log ) Simplify: ln a - 3 ln b + ln c 5
Expand: Simplify:
3) ln x y z ) 3 log x + 7 log y - 4 log z 3 4 5

36. Log Properties Proofs

37. Practice

38. Warm-up

39. Writing Exponential Functions to Predict Data
Pearson 7-1, Problem #5
y - y y
r = 2 1 1

40. Using Exponential Regression to Solve Problems
To make great coffee, Starbucks brews its coffee between 195 and
205. However, coffee is cool enough to drink at As one cup of
coffee cooled, its temperature was measured to show the
exponential model below. How long is it until the coffee is cool
enough to drink?
Time (min) 5 10 15 20 25 30
Temp ( F)
203
177
153
137
121
111
104

41. M&M Activity

42. Assessment Possibilities
pg. 474 #58 (Plus Real-World Questions)
Doubling Time Discovery (From curriculum
guide, change numbers from Paideia
problems)