Name Class Period. Secondary 1 Honors Unit 4 ~ Exponential Functions

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1 Name Class Period Secondary 1 Honors Unit 4 ~ Exponential Functions

2 Schedule for Unit 4 A-Day B-Day What we re doing Assignment What is due? Nov. 10 Nov : Graphing Exponential Functions 4-1 Nov. 14 Nov : Vertical Transformations of Exponential Functions Nov. 16 Nov. 17 Quiz on 4-1 & : More Vertical Transformations Nov. 18 Nov : Growth & Decay Nov. 22 Nov : Compound Interest Nov. 29 Nov. 30 Quiz on 4-4 & 4-5 Around the Room Activity / Start Review Start Review 4-5 Dec. 1 Dec. 2 Finish Review Review Dec. 5 Dec. 6 Unit 4 Test Review

3 NOTES 4-1: Graphing Exponential Functions Exponential Function: Example 1: Graph the function yy = 22 xx x y Asymptote:

4 Ex. 2: Graph the function yy = 33(22) xx x y Ex. 3: Graph the function yy = xx x y

5 Ex. 4: Graph the function yy = 22 xx x y Ex. 5: What would the graph of yy = ( 22) xx look like?

6 Homework Assignment 4-1: Graphing Exponential Functions Instructions - for problems 1-11: a) Make a table showing your work on a separate paper. Find values for x=-2, -1, 0, 1, and 2. b) Graph each exponential function. You must graph the 5 points found in your table. Use graph paper to make your graphs. Make sure to label your axes. c) State whether the function is increasing or decreasing, and if the function is above or below the asymptote. 1. yy = 4(2) xx 2. yy = xx 3. yy = 3(7) xx 4. yy = xx 5. yy = 2(5) xx 6. yy = xx 7. yy = 6(3) xx 8. yy = 1 2 xx 9. yy = (8) xx 10. yy = xx 11. yy = 5 xx 12. Try graphing this equation WITHOUT making a table. Use the initial value (starting point) and the common ratio. yy = 3(2) xx

7 NOTES 4-2: Vertical Transformations of Exponential Functions Using the parts to create the sequence in the table: Example 1: Given the functions ff(xx) = 2 xx and gg(xx) = 2 xx 3, complete the following table and use it to graph both functions on the same graph. y 20 Work ff(xx) = 2 xx gg(xx) = 2 xx How does the 33 affect the graph of the function? What has happened? x f ( x ) ( ) What is the y-intercept? : g x : f ( x ) ( ) Where is the asymptote? : g x : Are the functions increasing or decreasing? Are the functions above or below their asymptotes?

8 Example 2: Given the parent function ff(xx) = 1 3 xx and gg(xx) = 1 3 xx 5, which has been shifted vertically, fill in the table and complete the following questions. Work ff(xx) = 1 3 xx gg(xx) = 1 3 xx How does the 5 affect the graph of the function? What has happened? f ( x ) ( ) What is the y-intercept? : g x : f ( x ) ( ) Where is the asymptote? : g x : Are the functions increasing or decreasing? How can we tell without graphing? Are the functions above or below their asymptotes? How can we tell without graphing? Homework Assignment 4-2: Vertical Transformations of Exponential Functions Do the worksheet.

9 NOTES 4-3: More Vertical Shifts Identify the y-intercept: Identify the asymptote: Graph. Ex. 1: ff(xx) = 22(33) xx 44 y What is the y-intercept? Where is the asymptote? Are the functions increasing or decreasing? Are the functions above or below their asymptotes?

10 Graph. Ex. 2: ff(xx) = 55(22) xx y What is the y-intercept? Where is the asymptote? Are the functions increasing or decreasing? Are the functions above or below the asymptotes? Ex. 3: ff(xx) = xx + 55 y What is the y-intercept? Where is the asymptote? Are the functions increasing or decreasing? Are the functions above or below the asymptotes?

11 Example 4: For this problem you are given the parent function ff(xx) = 2(3) xx and a second function gg(xx) = 2(3) xx + 4 that has been shifted vertically. a) Create a table for both ff(xx) and gg(xx). b) Graph both ff(xx) and gg(xx) on the same graph. Use graph paper. Make sure to label your axes and draw the asymptotes. c) Answer the following questions. What is the y-intercept? ff(xx) : gg(xx) : Where is the asymptote? ff(xx) : gg(xx) : Are the functions increasing or decreasing? Are the functions above or below their asymptotes?

12 Identify the y-intercept and the asymptote for each function. Ex. 5: ff(xx) = 44(22) xx 11 Ex. 6: gg(xx) = 55(22) xx Ex. 7: hh(xx) = 55 xx 77 Ex. 8: ff(xx) = (33)xx + 11 Ex. 9: ff(xx) = 99(22) xx

13 Homework Assignment 4-3: More Vertical Shifts Instructions for #1 to #6: a) Make a table of values for x=-2, -1, 0, 1, 2 b) Graph. Make sure you label your graph and asymptote. c) Identify the y-intercept and asymptote for each graph. 1) ff(xx) = 2 xx + 3 2) yy = 1 5 xx 2 3) ff(xx) = 4(3) xx + 1 4) yy = xx + 3 5) ff(xx) = xx 6) yy = 7 1 xx For problems #7 to #10 you are given the parent function ff(xx) and a second function gg(xx) that has been shifted vertically. a) Create a table for both ff(xx) and gg(xx) on graph paper. b) Graph both ff(xx) and gg(xx) on the same graph. Use graph paper. Make sure to label your axis and draw the asymptotes. c) Answer the question: What is the y-intercept? Where is the asymptote? Are these functions increasing or decreasing? Are these functions above or below the asymptote? 7) ff(xx) = 5 xx, gg(xx) = 5 xx 2 8) ff(xx) = 1 4 xx, gg(xx) = 1 4 xx 1 9) ff(xx) = 1 5 xx, gg(xx) = 1 5 xx ) ff(xx) = (8) xx, gg(xx) = (8) xx + 4 Identify the y-intercept and asymptote of the function, without graphing. 11) ff(xx) = (6) xx 4 12) ff(xx) = 5(2) xx ) gg(xx) = 4 xx ) yy = 8(15) xx ) yy = 6(3) xx 1 16) yy = 5(6) xx ) h(xx) = 1 3 (2)xx 7 18) yy = 27(4) xx 14

14 NOTES 4-4: Growth & Decay Exponential Functions: Exponential Growth Functions: Growth Factor: Rate: Ex. 1: In 1996, there were 2573 computer viruses and other computer security incidents. The number of incidents increased by about 92% each year. a) Write an exponential model (equation) giving the number n of incidents t years after b) Use your equation to determine how many incidents were there in 2003?

15 Exponential Decay Functions: Decay Factor: Ex. 2: A new snowmobile costs $4200. The value of the snowmobile decreases by 10% each year. a) Write an exponential model (equation) giving the snowmobile s value y (in dollars) after t years. b) Estimate the value of the snowmobile after 3 years. Steps for writing an exponential growth or decay function:

16 Ex. 3: A ball is dropped from 8 feet above the ground. Each bounce is 80% of its previous height. a) Write an exponential model giving the ball s height h (in feet) after each bounce n. b) How high will the ball be after 5 bounces? Ex. 4: The algae population begins with 1000 in The algae population in a pond increases by 3% each year. a) Write an exponential model giving the number n of algae t years after b) How big will the population be in 2018? Ex. 5: The air slowly kills bacteria found on a surface. The bacteria population began at 3000 and decreases by 5% each minute. a) Write an exponential model giving the population p of bacteria after m minutes. b) What will be the bacteria population after 21 minutes?

17 Ex. 6: A number of bacteria, ff(tt), at any time tt, in hourse, can be estimated using the function ff(tt) = 3000(1.24) tt. a) What was the initial size of the bacteria colony? b) What is the growth/decay factor for the situation? c) Is the bacteria population exponentially decaying or growing? How do you know? d) At what rate is the population changing?

18 Homework Assignment 4-4: Growth & Decay For questions 1-3, do the following: a) identify the initial amount b) identify the growth or decay factor c) identify the annual percent increase or decrease 1. From 1997 to 2002, the number n (in millions) of DVD players sold in the United States can be modeled by nn = 0.42(2.47) tt where t is the number of years since Each March from 1998 to 2003, a website recorded the number y of referrals it received from Internet search engines. The results can be modeled by yy = 2500(1.50) tt where t is the number of years since The value of a car can be modeled by the equation yy = 24,000(0.845) tt where t is the number of years since the car was purchased. 4. Adella bought a car for $10,000. One year later, the car was worth $8,000. A year after that, the car was worth $6,400. a) Write an explicit equation for how much the car will be worth after n years. b) How much will the car be worth after 5 years (Round to the nearest hundredth)? 5. The Work-Out Gym sold 550 memberships in Since then the number of memberships sold has increased 3% annually. a) Write an explicit equation. b) How many members will there be in 2020? (Round to the nearest whole number.) 6. The number of people who own computers has increased 23.2% annually since In 1990, half a million people owned a computer. a) Write an explicit equation. b) Predict how many people will own a computer in (Round to the nearest whole number.) 7. Cami purchased a rare coin form a dealer for $300. The value of the coin increases 5% each year. a) Write an explicit equation b) How much will the coin be worth in 5 years? (Round to the nearest hundredth.) (continued on the next page)

19 8. In the years from 2010 to 2015, the population of the District of Columbia is expected to decrease about 0.9% annually. In 2010, the population was about 530,000. a) Write an explicit equation. b) What is the population expected to be in 2015? (Round to the nearest whole number.) 9. Leonardo purchases a car for 18,995. The car depreciates at a rate of 18% annually. After 6 years, Manuel offers to buy the car for $4,500. Should Leonardo sell the car? Explain. 10. Susan puts her $2,000 she saved from her summer job into a savings account. The account earns 1.6% interest each year. a) Write an explicit equation b) How much money will she have in 13 years? (Round to the nearest hundredth.) 11. List all of the growth functions in the table below. y = 0.3( 1.5) t y = 7( 0.53) t 65( 0.987) t y = 2001 y = 4.5( 2.58) x y = ( ) x 12. You are running a new city. a. Choose a starting amount for the population in your new city. b. Your city is growing by 300%. Write an explicit equation for your city. c. Which of the following statements is true about your city? A. Your city s population is doubling every year. B. Your city s population is tripling every year. C. Your city s population is quadrupling every year.

20 NOTES 4-5: Compound Interest For real world situations involving interest, use the model: A = P 1 + r n nt Where A= P= r= n= t= Example 1: You deposit $4000 in an account that pays 2.92% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency. a. Annually b. Quarterly c. Monthly

21 Example 2: Sally invests the same amount of money in three different bank accounts that earns 2.5% interest compounded at different rates. She starts by putting in $2,500. In which bank will Sally have more money after the given amount of time. Bank A: Compounded monthly for 2 years. Bank B: Compounded annually for 4 years. Bank C: Compounded quarterly for 3 years.

22 Homework Assignment 4-5: Compound Interest 1. Cami invested $6,000 dollars into an account that earns 10% interest compounded annually. a. Write an explicit equation for how much money she will have after tt years. b. How much money will Cami have in 6 years? Round to the nearest hundredth. 2. Sarah s saving account currently has $200. She earns 5% interest on her account compounded monthly. a. Write an explicit equation for how much money she will have after tt years. b. How much money will Sarah have after 6 months? Round to the nearest hundredth. 3. Paul invested $400 into an account with a 5.5% interest rate compounded monthly. a. Write an explicit equation for how much money she will have after tt years. b. How much will Paul s investment be worth in 8 years? Round to the nearest hundredth. 4. Theo invested $6,600 at an interest rate of 4.5% compounded monthly. a. Write an explicit equation for how much money he will have after tt years. b. How much will Theo s investment be worth in 4 years? Round to the nearest hundredth. 5. Paige invested $1200 at an interest rate of 5.75% compounded quarterly. a. Write an explicit equation for how much money she will have after xx years. b. How much will Paige s investment be worth in 7 years? Round to the nearest hundredth. 6. Brooke is saving money for a trip to the Bahamas that costs $ She puts $150 dollars into a savings account that pays 7.25% interest compounded quarterly. Will she have enough money in the account after 4 years? Explain. 7. Jin s investment of $4,500 has been losing its value at a rate of 2.5% each year. a. Write an explicit equation for how much money he will have after xx years. b. How much will Jin s investment be worth in 5 years? Round to the nearest hundredth. 8. Santos invested $1,200 into an account with an interest rate of 8% compounded monthly. James invested $1,500 into an account with an interest rate of 5% compounded quarterly. a. Write an explicit equation for how much money Santos will have after x years. b. Write an explicit equation for how much money James will have after x years. c. Who will have more money after 5 years? d. Who will have more money after 7 years? e. Who will have more money after 10 years?

23 ANSWERS TO SELECTED QUESTIONS: 4.1: : ff(xx): (0, 1), gg(xx): (0, 6) ff(xx): yy = 0, gg(xx): yy = 6

9. ff(xx): yy = 0, gg(xx): yy = 3 11. ff(xx): (0, 2), gg(xx): (0, 1) 13. 15.

25. ff(xx): yy = 0, gg(xx): yy = 2 27. below 29. ff(xx): (0, 3), gg(xx): (0, 2) 31. decreasing 4.

24 9. ff(xx): yy = 0, gg(xx): yy = ff(xx): (0, 2), gg(xx): (0, 1) ff(xx): yy = 0, gg(xx): yy = above 19. ff(xx): (0, 1), gg(xx): (0, 3) 21. decreasing ff(xx): yy = 0, gg(xx): yy = below 29. ff(xx): (0, 3), gg(xx): (0, 2) 31. decreasing 4.3: y-ints: ff(xx): (0, 1), gg(xx): (0, 1) asymptotes: ff(xx): xx aaaaaaaa, gg(xx): yy = 2 Increasing above

25 9. y-intercepts: ff(xx): (0, 1), gg(xx): (0, 2) asymptotes: ff(xx): xx aaaaaaaa, gg(xx): yy = 1 Decreasing Above 11. y-int: (0, -5), asymptote: y= y-int: (0, 16), asymptote: y= y-int: (0, 5), asymptote: y= y-int: 0, 7 1, asymptote: y= : 1a) 0.42, 420,000 dvd players 1b) c) 147% 3a) $24,000 3b) c) -15.5% 5a) ff(xx) = 550(1.03) xx b) ff(2020) = 964 mmmmmmmmmmmmmm 7a) ff(xx) = 300(1.05) xx 7b) ff(5) = $ No. (Your work and explanation must be shown.) 11. yy = 0.3(1.5) tt, yy = 4.5(2.58) xx, yy = 0.41(1.1) xx 4.5: 1a) yy = 6,000(1.1) tt 1b) $10, a) yy = tt 3b) $ a) yy = xx 5b) $1, a) yy = 4500(0.975) xx 7b) $3,964.93

Why? Exponential Growth The equation for the number of blogs is in the form 1 y = a(1 + r ) t. This is the general equation for exponential growth.

Why? Exponential Growth The equation for the number of blogs is in the form 1 y = a(1 + r ) t. This is the general equation for exponential growth. Then You analyzed exponential functions. (Lesson 9-6) Now Growth and Decay 1Solve problems involving exponential growth. 2Solve problems involving exponential decay. Why? The number of Weblogs or blogs