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?

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

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!)

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?

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?

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!)

Warm-up

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?

Graphing Exponential Functions Discovery: 1) Graph y = 2 on your calculator. x x + 4 2) Graph y = 2 on your calculator. (Remember to put x + 4 in parentheses.) x 3) Graph y = 2 + 4 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?

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)

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?

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

Warm-up

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

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 =

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) = (1 + ) 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) =

e Value The limit as n got bigger and bigger approaches 2.718281828... This is called the natural logarithm, or e. e is a NUMBER. What other letters represent numbers?

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?

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?

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?

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.

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

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)

Warm-up

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 243 2) log 216 3) log 19 4) log 2 3 6 3 1.04

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

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

Problem Task

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) 4 - 5 = 17 2) 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

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

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.

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.?

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 = 327 2) 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?

Warm-up

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 2) Simplify: ln a - 3 ln b + ln c 5 Expand: Simplify: 3) ln x y z 4) 3 log x + 7 log y - 4 log z 3 4 5

Log Properties Proofs

Practice

Warm-up

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

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 185 . 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

M&M Activity

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