Happy Wednesday Please do the following: Pick up the worksheet from the front table. HW #8: p 306 # 53, 57, odd, 91, 103, 107 Update: Test 12/ 16 and 12/17

Expectations I do not want to see your phone in my class. I do not care the reason, I DO NOT want to see it!!!! If you are on it, I will confiscate it until the end of the period. NO QUESTIONS! Please put it away in your backpack AND leave it there for the remainder of the class. I don’t care if there is only 5 minutes left of the class period. I WILL TAKE IT.

Expectations There is no cursing in my class. No throwing objects.

Office Hours! FLEX on Thursday!!!

Agenda 1)Review HW 2)Review Quiz 3)4.3: Exponential Functions 4)Practice Time!

Warm-Up! Tell whether the function shows growth or decay. What is the initial amount ( C value) and your growth factor ( a value) (a) p(x) = 5(1.2 x ) (b) w(x) = 10 (0.75 x ) (c) k(x) = 0.5 (2 x )

Review HW!

Unit 2 Quiz 1 Sit next to your partner: I only graded one quiz so make sure you know the mistakes that you and your partner made. Common Mistakes: #5) work backwards! #6) Read directions and know your transformations Students who need to take it: Period 7: Antonio, Jackie

Unit 2 Quiz 1 A: B: C: Below a 17.5, you can retake it to get a 17.5 If you retake it, it will NOT be a partner quiz! If you got a good grade because your partner helped you, then make sure you know this material for the test next week!

Learning Target I can: Define the number e. Solve exponential equations.

Calculator Expectations The next week we are going to use calculators. The resource manager is going to come up and get the calcs. – TABLE #1 ( CALCS #1-4) – TABLE #2 ( CALCS 5- 8) – TABLE #3 ( CALCS 9-12) – TABLE #4 ( CALCS 13-16) – TABLE #5 ( CALCS 17-20) – TABLE #6 ( CALCS 21-24) – TABLE #7 ( CALCS 25-28) 5 minutes before the bell rings, the resources manager is going to put the calcs away in the designated spot!!! NO ONE LEAVES MY CLASS UNTIL ALL CALCULATORS ARE PUT AWAY!!!!

Learning Objective(s) #2 By the end of this period you will be able to: ①Define e.

What is e?

11.3: The Number e The number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are … It is often called Euler's number after Leonhard Euler. e is the base of the natural logarithms (invented by John Napier). (It is called the natural base since you are able to take its derivative and you still receive the same function. ~ for those who were interested)

11.3: The Number e On your notes sketch y = e x. Answer the following questions on your notes:  What is the y- intercept?  What is the asymptote?  What is the domain?  What is the range?

11.3: The Number e The number e is used with natural logarithms. It is commonly used in business with investments or science for carbon dating or half-life. We will get to that more next week!

Learning Objective #3 By the end of this period you will be able to: ①Solve exponential equations

Learning Objective #3 By the end of this period you will be able to: ①Solve problems involving exponential growth and decay

11.2 Exponential Functions For real life scenarios, growth or decay is commonly represented by a constant percent increase or decrease using the following formula: In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor. Note: Rate is represented as a decimal, r = %/100.

Example 5 Example 5: You start a new job with an initial salary of $36,000 per year. Each year thereafter, you receive a 3% raise. Let S(t) be your salary t years after you start your new job. (a) Write the formula S(t). Is this a growth or decay? (b) What will your salary be after 10 years? (c ) When will your salary reach $72,000? (Use your graphing calculator to help you out ) You can graph it to help you figure it out. Write two equations on your graphing calculator! Next week, we will figure out how to do it algebraically

Whiteboard! Clara invests $5000 in an account that pays 6.25% interest per year during her first year of college. After 4 years of college, how much would she have in her account?

Group Work! You now have the time to work on the worksheet. We will go over Part I and Part 2 at _____________.

Many financial models use exponential functions. Let’s apply what we have learned to what everyone loves: $$$$$$$$$$$$$$$ There will be a lot of formulas. I am not asking you to memorize them. I am, however, asking you to know which one to use and how to use it!

4.7: Financial Models Compound Interest o A = final amount ($) o P = initial amount or principal o r = annual interest rate(%) o n = number of times the interest is compounded per year o t = time in years

4.7: Financial Models Common Compounding Periods o n is the number of times the interest is compounded per year  annually, n =  semiannually, n =  monthly, n =  quarterly, n =  daily, n =

Example 1 Example 1: Find the Future Value of a Lump Sum of Money Use the compound interest formula to calculate the amount of money you would have after 5 year if you invest $300 at an annual rate of 4.3% compounded: (a) Annually(c) Monthly (b) Quarterly(d) Daily (e)What do you notice as you increase n? Hint: When putting this in your calculator remember parentheses!!!

4.7: Financial Models Assume that you have $1 and it earns 100% annual interest. This table shows the growth factor for each of the compounding frequencies listed.

4.7: Financial Models Continuously Compounded Interest o A = Pe rt, where A is the total amount, P is the principal, r is the annual interest rate (%), and t is the time in years. o Continuously Compounded interest is when interest (a fee) is added to a deposit or loan, so that, from that moment on, the interest that has been added also earns interest. o It is different from the compounded interest formula

Example 2 Example 2: Continuous Compounding How much money will your credit card have after loaning you $5000 at 11.9% interest compounded continuously for 4 years? How much of that is just interest?

Example 3 Congrats! You just won the lottery! You are smart and going to invest your money for 30 years. However, you are debating on whether to invest your $50,000 winnings into a money market that earns 8.5% interest compounded quarterly or into a savings account that earns 7.9% interest compounded continuously. Which option would you choose and why?

Whiteboards Compare the balance after 25 years of a $10,000 investment earning 6.75% interest compounded continuously to the same investment semiannually. Which option would you choose?

11.3: The Number e Table- Talk: Discuss the difference between interest compounded continuously and interest compounded monthly.

Group Work! 1) You are saving for college! You invested $2000 today and left it in your money market account until December How much money would you have if the annual rate is 5% and it is compounded monthly? 2) Andrew invested $500 to save for a new car. If he left the money in an account which compounds continuously, how much would he have in 20 years?

In Summary…

WhiteBoards If you have an investment compounded daily OR you have an investment compounded continuously, does it matter which formula that you use?

Creation! Take out a half sheet of binder paper. Write your name & table # in the upper right hand corner. Create your own word problem ( exponential growth/decay) The most creative will be used on the next quiz, and you will get extra credit!