Exponential Functions: (pp 296; 328) A function of the form y=a(b x ) were a 0 and 0<b<1 or b>1

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1 TABLE OF CONTENTS I. Day 1 Unit 5 Exponential Functions Orientation Day 2 Exponential Growth vs. Exponential Decay Day 3 Unit Five Lesson One Investigation One (5.1.1) Exponential Growth Day 4 Unit Five Lesson One Investigation One (5.1.1) Exponential Growth cont. Day 5 Day 6 Unit Five Lesson One Investigation Two (5.1.2) Compound Interest Day 7 Unit Five Lesson One Investigation Three (5.1.2) Compound Interest cont.. Day 8 Unit Five Lesson One Investigation Three (5.1.3) Compound Interest Day 9 Unit Five Lesson One Investigation Three (5.1.3) Compound Interest cont. Day 13 Unit Five Lesson Two Investigation One (5.2.1) More Bounce to the Ounce Day 14 Unit Five Lesson Two Investigation Two (5.2.2) Medicine and Mathematics Day 15 Unit Five Lesson Two Investigation Two (5.2.2) Medicine and Mathematics cont Day 16 Unit Five Lesson Two Investigation Four (5.2.4) Properties of Exponents II Day 17 Unit Five Lesson Two Simple Interest Exercise Day 18 Unit Five Lesson Two Compound Interest Exercise Day 19 Unit Five Assessment Review Day 20 Unit Five Assessment Review Day 21 Unit Five Exam

2 Day 1 Unit 5 Exponential Functions Benchmarks: 2.1a Model nonlinear real world situations with equations/inequalities. 2.3a Solve problems using graphs, tables, and algebraic methods. Terms: Exponential Expressions: An algebraic expression in the form of b n, where b and n are real numbers or variables. The number b is called the base of the exponential expression, and n is called the exponent or the power. Exponential Functions: (pp 296; 328) A function of the form y=a(b x ) were a 0 and 0<b<1 or b>1 Example: Exponential Expressions 2 2 = 2*2 = b 2 = b*b = = 2*2*2 = b 3 = b*b*b = = 2*2*2*2 = b 4 = b*b*b*b = End of Day Day 2 Exponential Functions cont. Notice: The variable x is an exponent. As such, the graphs of these functions are not straight lines. In a straight line, the "rate of change" is the same across the graph. In these graphs, the "rate of change" increases or decreases across the graphs. Observe how the graphs of exponential functions change based upon the values of a and b: Example: When a > 0 and the b is between 0 and 1, the graph will be decreasing (decaying). For this example, each time x is increased by 1, y decreases to one half of its previous value. Example: When a > 0 and the b is greater than 1, the graph will be increasing (growing). For this example, each time x is increased by 1, y increases by a factor of 2. X Y= X Y=

3 10 10 Such a situation is called Exponential Decay. Such a situation is called Exponential Growth End of Day JAN05 Day 3 Unit Five Lesson One Investigation One (5.1.1) Exponential Growth Read Page 290 to understand the objective and desired outcomes of the Investigation. Essential Question- What are the basic patterns of exponential growth in variations of the Pay it Forward Process? 1. Follow Problem 1 on page 292 a. Graph b. c. 2. Follow Problem 2 on page 292 a. Tree Graph b. Table (emulate table in problem 1) Graph c. d. 3. Follow Problem 3 on page 292 a. NOW Next Rule Problem 1 NOW Next Rule Problem 2 b.

4 c. 4. Follow Problem 4 on page 292 a. b. c. d. TURN INTO INSTRUCTOR WHEN COMPLETE Day 6 Unit Five Lesson One Investigation Two (5.1.2) Getting Started Read Page 294 to understand the objective and desired outcomes of the Investigation. Essential Question- -What are the NOW-NEXT and y=? rules for basic exponential functions? -How can those rules be modified to model other similar patterns of change? 1. Follow Problem 1 on page 295 a. How many bacteria would exist after 8 hours (32 quarters) if the infection continues to spread as predicted b. 3 Number Qtr hrs Number Bacteria NEXT= Number Qtr hrs Number Bacteria NEXT= Number Qtr hrs Number

5 Bacteria NEXT= Number Qtr hrs Number Bacteria NEXT= Day 7 Unit Five Lesson One Investigation Two (5.1.2) Getting Started continued.. 4 Given: Now 4 people pay it forward 4a. Write the Next/Now Statement Next = 4b. Write the Rule N= 4c. How would the NOW-NEXT BE different if we started with 5 instead of four 4d. 5 Calculator problem: 5a. 5b. 6 X Y X Y X Y X Y 6a. 6b. 7: Given bacteria starts at 64 and doubles every hour 7a. What rule predicts the number at anytime x hours? 7b. What would it mean to calculate values for negative x? 7c. What would you expect for x= -1? -2?

6 -3? -4? 7d.explain 8 Explain problems a, c, e below End of Investigation Getting Started DAY 8 Investigation Compound Interest (Page 298) Use a calculator and record the following in your note book. What is going on: 1.1^2=? ^2=? ^2=? ^2=? What is going on? Essential Question- -How can you represent and reason about functions involved in investments paying compound interests? Given: You won the lottery and are given a choice to accept 10,000 dollars now or wait and collect in ten years. If you can invest the 10,000 in an 8% money market now, what will be the better of the two choices? 1.) Discuss the choices and write an explanation of your decision 2.) Write the rules a. For the next year and after (NOW-NEXT) 1. ANS. NEXT= NOW + NOW(.08) STARTING AT 10,000 OR NEXT = NOW(1+.08) STARTING AT 10,000---factoring out now OR NEXT = NOW(1.08) STARTING AT 10,000 b. After any year 1. ANS. Y = 10,000(1.08 X ) 3.) Use the rule from number 2b. to determine the value of the after 10 years 4.) Answer the following a. Describe the pattern ANS. The pattern is the growth starts at 800 per year and increases until nearly 1600 after ten years. b. Why isn t the change in CD the same every year?ans. The yearly interest is compounded, meaning last year s interest is included in the Next years calculation. c. How is the pattern of increase in CD balance shown in the shape of a graph or function? ANS. The graph gently curves upward and the curve slowly becomes steeper. d. 5.) Solve the following using the calculator table sub-routine

7 6.) Solve the following _ a. Given earning 4% annual interest compounded yearly b. Given 5000 earning 12% annual interest compounded yearly The following table illustrates each year for 10,000 at 8% Missed funeral 10, Initial Dollars 8% Year 8% percent Compounded 0 8% 8% 1 10, , , , , , , , , , , , , , , , , , , , , , , , , four days for a

8 25JAN11 DAY 13 INVESTIGATION More Bounce to the Ounce (page 323) Essential Question: What mathematical patterns in tables, graphs, and symbolic rules are typical of exponential decay relations? Do problems 1 only. For problem 2 explain only why the data will be the same or different then the calculated results? 1.) Fill in table if bounce is 2/3 of previous bounce. Bounce # Rebound Ht a.) How does the rebound height change from one bounce to the next? b.) Write the NOW/NEXT Statement c.) Write the Rule: y= d.) How will the data table, plot, and rules for calculating the height change if the ball drops first from 15 feet? 2. Explain why the theoretical data will be different or the same End of Investigation

9 DAY 14 Review Quiz Questions INVESTIGATION Medicine and Mathematics (page 326) Essential Question: How can you interpret and estimate or calculate values of expressions involving fractional or decimal exponents? How can you interpret and estimate or calculate the half-life of a substance that decays exponentially? 1. According to the graph on page 327 what appears to be the halflife?. 2. Answer question 2 on page 327 a. Graph the equation given. b. What do 10 and.95 represent? c. What percent is used up every minute? 3. Write the NOW/NEXT Rule 4. Answer problem 4 on page 327/328 a. 9.3 b. 8.0 c Answer problem 5 on page 328 a. Fill in the table Time (min) 19.5Units in blood b. Compare the data with your graph c. Estimate the following values and solve for x i. X is about 31.4 ii. X is about 4.3 iii. X < about Answer question 6 on page 328 a. At 15 units b. At 20 units c. At 25 units d. Explain the result

10 Day 15 Investigation Medicine and Mathematics cont Daily Starter Solve for Y(x) when x=4 Continue and finish Investigation End Investigation Begin Homework Page 338 #3,4 Problem 3 Page 338 Given the following: The steroid cyprionate is illegal in athletics and 90% will stay in your system one day after injection. 90% of the remaining will be in your system one day later and so on In this scinerio an athlete injects 100-mg of cyprionate. 1. Make a table showing the amount of drug in the system after every day for one week Time days Steroid % Plot Data and describe the pattern Describe pattern:

11 3. Write two rules (NOW/NEXT and y= ) for the problem. NEXT= y= 4. Use one rule to find steroids left after.5 days= 8.5 days= 5. Estimate the half-life 6. How long until steroid is less than 1% of its original strength? Penicillin The most famous antibiotic drug is penicillin. After its discovery in 1929, it became known as the first miracle drug because it was so effective on fighting serious bacterial infections. Drugs act somewhat differently on each person. But on average, a dose of penicillin will be broken down in the blood so that one hour after injection only 60% will remain active. Suppose a patient is given an injection of 300 milligrams of penicillin at noon. a. Write the rule in the form of y=a(b x ) that can be used to calculate the amount of penicillin remaining after any number of hours. b. Use your rule to graph the amount of penicillin in the blood from 0 to 10 hours. Explain what the pattern of that graph shows about the rate at which active penicillin decays in the blood. c. Use your rule to develop a table showing the amount of active penicillin that will remain at quarter hour intervals from noon to 5 p.m. i. Estimate the half life ii. Estimate the time it takes for an initial 300mg dose to decay so that only 10 mg remain active. d. If 60 % of a penicillin dose remains active one hour after an injection, what percent has been down in the blood.

12 31Jan2011 Day 16 Investigation Properties of Exponents Essential Question: What exponent properties provide shortcut rules for calculating powers of fractions, quotients of powers and negative exponents Find values of x and y that make the statement true. 1. Solve problem on page 332 a. b. c. d. 2. Examine problem 1 a. What pattern do you see? b. Don t answer c. What are some common problems when doing these problems? 3. Solve problem 3 on page 332 a. Z= b. Z= c. Z= d. x= e. x= y= f. Z= g. Z= h. Z= 4. Examine problem 3 a. What pattern do you see? b. Don t answer c. What are some common problems when doing these problems? Solve problem 6 on page 333: Given Insect growth is based on the formula: p=48(2^x). 7. Write an equivalent fraction that does not use exponents at all. a. b. c. d. e. f. g. h.

13 8. Examine the results of your work in Problems 6 and 7. a. How would you describe the rule defining negative integer exponents in your own words? b. What are some common errors in evaluating negative integer exponents? End of Investigation Check your Understanding Day 17 Simple Interest Exercise Simple Interest Equation A=total amount earned including principle P= initial amount invested r= rate of interest t = # of years Essential Question: What is Simple Interest? What is Compound Interest? What are the differences between the two? Simple Interest Whenever you borrow money, you pay a usage fee. That fee is called interest: Interest (I) = the amount charged for the use of borrowed money. The amount of interest you pay is based on three elements: the amount you borrow, the interest rate, and the length of time the money is borrowed for. The terminology for these elements is as follows: Principle (P): the amount borrowed Interest Rate (r): annual percentage of the principle that is charged as a fee Term (t): length of time the money is borrowed

14 When it is time to pay back the money, you are required to pay the principle plus the amount of interest that has accumulated. This is called simple interest and it is typically used for very shortterm borrowing or investments. The formula is as follows: Interest = Principle * rate * time (I=P*r*t) Example: If you borrow 1000 for five years at an interest rate of 10%, the amount of interest you pay is: I = P*r*t I= 1000*0.10*5 I= 500 The cost of borrowing 1000 for five years at 10% interest is 500. The total amount (A) due at the end of five years is principle + interest: A = P + I A = = 1500 An efficient way to calculate the total amount owed: A = P * (1+ rt) A = 1000 * ( *5) A = 1000 * 1.5 A = 1500 When you borrow money, you pay interest but when you invest money, you earn interest. An investment is really a case where you lend your money to someone else and they pay you interest. The same equations apply when calculating simple interest that is earned except now principle is the amount invested and interest is the amount earned. Example: If you put 5000 in a savings account that pays 2% annual interest, the amount of money you will have at the end of the year is: A = P * (1+ rt) A = 5000 * ( *1) A = 5000 * 1.02 A = 5100 The interest earned is 100. With simple interest you can set up an interest table that shows how much interest is accumulated over time. If we use the same example of 5000 in a savings account that earns 2% interest annually, the interest table looks like this: Term Interest Total (Yrs) After 5 years the savings account would have Suppose you want to figure out how long it would take to have 7250 in your savings account. You could create an interest table like the one above but a much better method is to use algebra and rearrange the simple interest equation to solve for time (t).

15 Two methods: 1. Calculate when the account equals 7250 A = P * (1+ rt) t=[1-(a/p)] / r t=[1-(7250/5000)] / 0.02 t=22.5 years 2. Calculate when 2250 interest has been earned (I = A-P = ) I = Prt t = I/(Pr) t = 2250/(5000*0.02) t = 2250/(100) t = 22.5 years Now suppose you want to know what interest rate would be needed to earn 7250 on a 5000 investment in 15 years. Two methods: 1. Calculate the rate that generates 7250 total A = P * (1+ rt) r=[1-(a/p)] / t r=[1-(7250/5000)] / 15 r= 0.45/15 r=0.03 = 3% 2. Calculate the rate that generates 2250 interest (I = A-P = ) I = Prt r = I/(Pt) r = 2250/(5000*15) r = 2250/75000 r = 0.03 = 3%

16 Student Worksheet The amount of interest you pay is based on three elements: the amount you borrow, the interest rate, and the length of time the money is borrowed for. 1. The amount charged for the use of borrowed money is called: a. Principle b. Term c. Interest d. Rate 2. The amount of an original investment is called: a. Principle b. Term c. Interest d. Rate 3. The formula for Simple Interest is: a. P=itr b. I=Prt c. r=i/pt d. Both b and c 4. The three elements used to calculate simple interest are,,. 5. How much interest does a 10,000 investment earn at 5.6% over 18 years? 6. Susan borrows 8650 to buy a used car and is charged 4.5% interest. If the term of her borrowing is 5 years, how much interest does she pay in total? 7. Henry invests 5000 in a mutual fund with an annual interest rate of 7.5%. He also has a 4-year, 10,000 loan at 3.75%. When will the amount of interest earned on the mutual fund be equal to the amount of interest paid on the loan? 8. If Sheila paid in interest on a 5 year loan of 5,800. What was the interest rate?

17 Day 18 Compound Interest Exercise Exponents, Compound Interest A special case of exponential functions is compounding or the increasing of money over time such that every time the invested money grows, the new amount is used to calculate interest. Compounding interest formula: A=total amount earned including principle P= initial amount invested r= rate of interest n= # of times compounded (per year) t = # of years 1. If Jack invests 5,000 in an account at 6% interest, a. Compounded annually, what is his investment worth in 5 years? b. Compounded monthly, what is his investment worth in 5 years? c. How many weeks are there in a year?? d. What is the answer if we compound the interest every week for five years? 2. Grandpa Henry isn't sure which investment is better. Should he invest in a fund which pays 3.7% compounded monthly or a fund which pays 6.0% compounded annually? What is the amount of both if 1000 is invested for a total of 10 years? 3. Aunt Hildegarde likes free gifts so when her bank gave away toasters, she invested 2,500 in an account that is compounded monthly at 3.2%. Unfortunately, Aunt Hildegarde was somewhat senile and she forgot about the account and when she died, you inherited it. If the money was untouched for 38 years, how much did you inherit?