Unit 3 Day 8. Exponential Point-Ratio Form

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1 Unit 3 Day 8 Exponential Point-Ratio Form

2 Warm Up Time (minutes) 2. Without using your calculator, would you say that the data appears to e more linear or exponential? Explain. 3. Explain the steps you take to enter the data in the calculator and find a regression equation. Consult a neighor if you need some help. Temperature ( o F) Exponential Regression When handed to you at the drive-thru window, a cup of coffee was 200F. Some data has een collected aout how the coffee cools down on your drive to school Would you say that the data has a positive, negative, or no correlation? Explain. 4. Calculate the linear regression and use it to predict how cool the coffee will e in 5 minutes. 5. Calculate the exponential regression and use it to predict how cool the coffee will e in 5 minutes. 6. Which regression do you think provides a more accurate prediction? Explain your 2 reasoning.

3 Warm Up Time (minutes) Temperature ( o F) Exponential Regression When handed to you at the drive-thru window, a cup of coffee was 200F. Some data has een collected aout how the coffee cools down on your drive to school Would you say that the data has a positive, negative, or no correlation? Explain. Negative 2. Without using your calculator, would you say that the data appears to e more linear or exponential? Explain. Exponential ecause of a common r 0.95 ratio not a common difference Explain the steps you take to enter the data in the calculator and find a regression equation. Consult a neighor if you need some help. y 200(0.95) x 4. Calculate the linear regression and use it to predict how cool the coffee will e in 5 minutes. y 8.82x Find y (5) 66.5 F 5. Calculate the exponential regression and use it to predict how cool the coffee will e in 5 minutes. x y 200(0.95) Find y (5) 92.7 F 6. Which regression do you think provides a more accurate prediction? Explain your reasoning. 3 Exponential ecause of a common ratio is consistent, not a common difference

4 Homework Answers Packet p. 2 & 3 Population a. y = 3(.07) x. x represents the numer of years since 990 Year Population million mil mil mil mil c. a = 3 million, it represents the initial population of Florida in 990. d. =.07, it represents the ase of the function, what we multiply y at each iteration. 4

5 2 Healthcare a. y = 460(.08) x Homework Answers Packet p. 2 & (.08) 27 = $ c. 994 (Be Careful! Do intersection in calculator THEN x = = , which is still in 994) d (Be Careful! Do intersection in calculator THEN x = = , which is still in 2003) 3 Half-Life y = 2(.5) x/8 **Be Careful! Half Life of 8 days means exponent must e x/8 so the exponent determines the numer of times we have halved!! Y = 2(.5) (4/8) = 3.57 millicuries 5

6 4 Savings Homework Answers Packet p. 2 & 3 a. 500(.065) 8 = $ = 500(.065) x and find intersection -> years c. 500(.080) 8 = $ So, the difference is $ d. y = 500(+(.065/2)) x. 8 years is 26 months. So 500(+(.065/2)) 26 = $ The difference is $

7 Homework Answers Packet p. 2 & 3 5 Health a. NEXT = NOW * (0.959), START = 6.5. y = 6.5(0.959) x c. 6.5(0.959) (-0) = gal 6 Washington D.C. a. 604,000 people..8% decay, ecause =.08 c. NEXT = NOW * (.08), START = 60,4000 d (.982) ,742 people e. Approximate year

8 Heads up aout tonight s Homework: Packet p. 4 AND Finish Notes p

9 Notes p.2 Point-Ratio Form 9

Point Ratio Form When you ring a ell or clang a pair of cymals, the sound seems loud at first, ut the sound ecomes quiet very quickly, indicating exponential decay.

10 Point Ratio Form When you ring a ell or clang a pair of cymals, the sound seems loud at first, ut the sound ecomes quiet very quickly, indicating exponential decay. This can e measured in many different ways. The most famous, the deciel, was named after Alexander Graham Bell, inventor of the telephone. We can also use pressure to measure sound. While conducting an experiment with a clock tower ell it was discovered that the sound from that ell had an intensity of 40 l/in 2 four seconds after it rang and 4.7 l/in 2 seven seconds after it rang.. Using this information, what would you consider to e the independent and dependent values in the experiment? x = time y = intensity 2. What two points would e a part of the exponential curve of this data? (4, 40) and (7, 4.7) 3. What was the initial intensity of the sound? 4. Find an equation that would fit this exponential curve. Let s talk aout how to 0 find these!

11 Point Ratio Form You can find this equation using exponential regression on your graphing calculator, ut what if you don t have a graphing calculator? You may recall learning how to write equations for linear information using point-slope form. This is a little it different, oviously, ecause we re working with an exponential curve, ut when we use point-ratio form, you should see some similarities. y y Point-Ratio Form x x Where (x, y) and (x, y ) are ordered pairs that can e used to find the value of, the common ratio. OR Where (x, y ) an ordered pair and the common ratio,, can e used to write an explicit equation for the scenario or identify other possile values for the scenario.

12 Point-Ratio Form Where (x, y) and (x, y ) are ordered pairs that can e used to find the value of, the common ratio. To find the value of Step : Identify two ordered pairs. Laeling the points can help -> x y x y Step 2: Sustitute the values into point-ratio form. Step 3: Simplify the exponents Step 4: Divide to get the power alone. Step 5: Use the properties of exponents to isolate (40) (40) y y Let s look ack at the Bell prolem to see how to solve for an exponential formula y hand using the following steps! Here s what we have so far! (7, 4.7), (4, 40) (7 4) Now, we need to find the initial value, a, to finish the formula!! 2 Next slide -> y a(.4898) x x x Round to 4 places for the formula!!

13 Point-Ratio Form Where (x, y) and (x, y ) are ordered pairs that can e used to find the value of, the common ratio. Here s what we have so far! To find the initial value, a Step : Identify one ordered pair and the value. (4,40),.4898 y y Let s look ack at the Bell prolem to see how to solve for an exponential formula y hand using the following steps! y a(.4898) x x x Step 2: Sustitute the values into exponential form. Step 3: Simplify to solve for a. Rememer, we rounded to 4 places for the formula. BUT to get a more exact a value, we should use the ANS utton on the calculator!! a(.4898) 4 40 a(.4898) 4 4 (.4898) (.4898) a y (.4898) x Final Point Ratio Formula!

14 Want to watch this example again? Watch it on YouTue! 4

15 Notes p Guided Practice Point-Ratio Form 5

16 Notes p Guided Practice Some of the ANSWERS are on next slides 6

17 ) The intensity of light also decays exponentially with each additional colored gel that is added over a spotlight. With three gels over the light, the intensity of the light was 900 watts per square centimeter. After two more gels were added, the intensity dropped to 600 watts per square centimeter. x x a) y y Write an equation for the situation. Show your work. (3,900) (5, 600) (53).865 (3,900) 900 a(. 865) a y 653.4(.865) x ) Use your equation to determine the intensity of the light with 7 colored gels over it. 7 y 653.4(.865) IninitialValue 7

18 2) Using the points (2, 2.6) and (5, ). y y x x a) Find an exponential function that contains the points. Show your work. (2,2.6) (5, ) (52) (2,2.6) 2.6 a(.5) a 5.6 IninitialValue 2 ) Rewrite the function so that the exponent of the function is simply x. y 5.6(.5) x 8

19 3) When the rown tree snake was introduced to Guam during World War II y the US military it devastated the local ecosystem when its population grew exponentially. If snake was accidentally rought to Guam in 945 and it was estimated that there were 625 snakes in 949, write an equation for the situation and use it to predict how many snakes there were in 955. In your equation, let x = numer of years after 945. Show your work. x x (0,) (4, 625) 625 y (625) 5 x (40) 4 5 y y 0 y ( 5) 9,765,62 5 snakesin0 years 9

20 4) (0, 76.9) (0, ) The water hyacinth that was introduced to North Carolina from Brazil ended up clogging our waterways and altering the chemistry of the water. We re not certain exactly when the water hyacinth was introduced, ut there were 76.9 square miles of water hyacinths in 984. Ten years later, there were square miles of watch hyacinth. Using this information, calculate when there was less than 0. square miles of this invasive plant in x x our waterways. y y 0 0 (00) (0,76.9) 76.9 a( 2.005) a IninitialValue y 76.9(2.005) x (2.005) x Put in calculator to solve! Year974 20

21 5) You recently discovered that your great aunt Sue set up a ank account for you several years ago. You ve never received ank statements for it until recently. The first month you received a statement for the account, you had $ in the account. One month later, the account had $ What is the approximate percentage of monthly interest that this account is earning each month? a) What is the approximate percentage of monthly interest that this account is earning each month? (,234.56) (2,296.29) r (2).05 r r. 05 y y ) In what month will your account have douled? x x 5% interest 2

22 Finishing Notes p is part of the HW, so we can discuss more questions later 22

23 Notes p. 25 An application of point-ratio form: finding the monthly rate of depreciation. Read the following question then we ll discuss it! Steps ) Use the info in the prolem to get 2 coordinate pairs. (0, 3000) use coefficient = initial value (, 0660) use equation 3000(0.82) to find the value after year 2) Since they want a monthly rate, convert the ordered pairs to months. (0, 3000) (2, 0660) 2 months = year 3) Use point-ratio form to find the value THEN the rate! (0,3000) (2, 0660) r (20) 2

24 You Try! The value of a truck can e modeled y the function V(t) = 6,000(0.76) t, where t is the numer of years since the car was purchased. To the nearest tenth of a percent, what is the monthly rate of depreciation? 2.3%per month 24

25 ) Simplify Unit 3 Practice 5 72x y 2 343x y x y 3 9y 2) Solve 2x 4 x 6 x 4 3) Solve 4 3 5(2x 4) 80 x 2 25

26 Unit 3 Practice (continued) 4) Sr-85 is used in one scans. It has a half-life of 34 hours. Write the exponential function for a 5 mg sample. Find the amount remaining after 86 hours. Show work algeraically. 5) Write exponential function given (2, 8.25) and (5, 60.75). Show your work algeraically. 6) A plant 4 feet tall grows 3 percent each year. How tall will the tree e at the end of 2 years? Round your answer to the hundredths place. 7) A BMX ike purchased in 990 and valued at $6000 depreciates 7% each year. Next = Now *.93, Start = 6000 a) write a next-now formula for the situation ) write an explicit formula for the situation c) how much will the BMX ike e worth in 2006? d) in what year will the motorcycle e worth $3000? mg 5.70 ft y x 999 y 6000(.93) x $,878.79

27 Tonight s Homework: Packet p. 4 AND Finish Notes p

28 More Notes p answers 28

29 6) When the wolf population of the Midwest was first counted in 980, there were 00 wolves. In 993, that population had grown to 300 wolves! Assuming the growth of this population is exponential, complete each of the following. a) Write an explicit equation to model the data. Show your work! (0,00) (3, 300) (30) x = years after 980 y 00(.3023) x ) Write a recursive (NOW-NEXT) equation for the data. Next = Now *.3023, Start = 00 29

30 6) When the wolf population of the Midwest was first counted in 980, there were 00 wolves. In 993, that population had grown to 300 wolves! Assuming the growth of this population is exponential, complete each of the following. c) Time Since 980 (in years) Estimated Wolf Population ,00 d) Identify the theoretical and practical domain of this situation. Explain how you decided upon your answers. Theoretical domain: All real #s Practical Domain: x 0 30

31 7) A 2008 Ford Focus cost $5452 when purchased. In 203 it is worth $8983 if it s in excellent condition. a) Assuming the value of the Focus depreciates y the same percentage each year, what is that percentage? x = years after 2008 (0,5452) (5, 8983) (50) ) Write an explicit function for the value of the Ford Focus. y 5452(0.8972) x c) If the owner of the car plans on selling it in 208, how much should she expect to get for it? r r r 0.28% decrease y 5452(0.8972) $

Exponential Equations

Exponential Equations Recursive routines are useful for seeing how a sequence develops and for generating the first few terms. But, as you learned in Chapter 3, if you re looking for the 50th term, you