Exponential Functions

1. Exponential Functions The basics, solving, and applications Mr. Morrow 2/5/2013 – 2/7/2013

2. - Warm Up - • Suppose you want to purchase a brand new Audi R8. They retail at about $120,000. You are given three options to approach how you will save for this car. Which method would get you to the desired amount fastest? • Set aside $60,000 the first month. Every month there after set aside $5,000. • Set aside $60,000 the first month. The second month set aside $500. Every month there after double what you set aside the previous month. • Set aside $60,000 the first month. Every month there after set aside half of what you did the previous month.

3. - Warm Up : Worked Out - • Suppose you want to purchase a brand new Audi R8. They retail at about $120,000. You are given three options to approach how you will save for this car. Which method would get you to the desired amount fastest? • Set aside $60,000 the first month. Every month there after set aside $5,000. • Month 1 – 60,000 • Month 2 – 60,000 + 5,000 = 65,000 • Month 3 – 65,000 + 5,000 = 70,000 • … • Month 12 – 110,000 + 5,000 = 115,000 • Month 13 – 115,000 + 5,000 = 120,000

4. - Warm Up : Worked Out - • Suppose you want to purchase a brand new Audi R8. They retail at about $120,000. You are given three options to approach how you will save for this car. Which method would get you to the desired amount fastest? • Set aside $60,000 the first month. The second month set aside $500. Every month there after double what you set aside the previous month. • Month 1 – 60,000 • Month 2 – 60,000 + 500 = 60,500 • Month 3 – 60,500 + 1,000 = 61,500 • Month 4 – 61,500 + 2,000 = 63,500 • Month 5 – 63,500 + 4,000 = 67,500 • Month 6 – 67,500 + 8,000 = 75,500 • Month 7 – 75,500 + 16,000 = 91,500 • Month 8 – 91,500 + 32,000 = 123,500

5. - Warm Up : Worked Out - • Suppose you want to purchase a brand new Audi R8. They retail at about $120,000. You are given three options to approach how you will save for this car. Which method would get you to the desired amount fastest? • Set aside $60,000 the first month. Every month there after set aside half of what you did the previous month. • Month 1 – 60,000 • Month 2 – 60,000 + 30,000 = 90,000 • Month 3 – 90,000 + 15,000 = 105,000 • Month 4 – 105,000 + 7,500 = 112,500 • Month 5 – 112,500 + 3,750 = 116,250 • Month 6 – 116,250 + 1,875= 118,125 • Month 7 – 118,125 + 937.5 = 119,062.5 • Month 8 –119,062.5 + 468.75 = 119,531.25 • … • Wait a minute? Will we actually reach 120,000 this way?

6. - Warm Up : Worked Out - • Suppose you want to purchase a brand new Audi R8. They retail at about $120,000. You are given three options to approach how you will save for this car. Which method would get you to the desired amount fastest? • Set aside $60,000 the first month. Every month there after set aside half of what you did the previous month. • Wait a minute? Will we actually reach 120,000 this way? • Yes, but not through exhaustion… • What do you think??

7. - Solution - • Suppose you want to purchase a brand new Audi R8. They retail at about $120,000. You are given three options to approach how you will save for this car. Which method would get you to the desired amount fastest? • Set aside $60,000 the first month. Every month there after set aside $5,000. • 13 months • Set aside $60,000 the first month. The second month set aside $500. Every month there after double what you set aside the previous month. • 8 months • Set aside $60,000 the first month. Every month there after set aside half of what you did the previous month. • Approaching

8. - Exponential Functions - • Today we will be examining the follow for exponential functions: • Law of Exponents Review • Basic form of equation • Comparing with other expressions • Growth/ Decay • Translations • Dilations • The General Form • Solving Exponentials Equations How do you feel about higher level math on exponential functions? Yea…we’ll hold off on that

9. - Laws of Exponents (Review) - • If ,1 or -1 then bx = by if and only if x = y • b0 = 1 • (ab)x = axbx • If , a >0, and b > 0 then ax = bx if and only if a = b • 9. (bx)y = bxy

10. - Laws of Exponents (Review) - 3. 4. 5. Adding like radicals:

11. - Basic Form - The exponential function takes the form, Such that x is a real number greater than 0 but not equal to 1 Just as you have done with previous courses, the variable that you will be changing (independent) is x (now the exponent). So no how do the following compare?

12. f(x) = 2x f(x) = x2 (4, 16) (2, 4)

13. Properties of The function increases for all values of x *Never changes directions The function is always positive *All y-values above x-axis *Implies x-axis is an asymptote When x < 0 then 0 < y < 1 x = 0 y = 1 and, x > 0 then y > 1

14. Now graph the following and compare: -Exponential Growth-

15. - Exponential Decay - Thus far we’ve seen Let’s think about This is in fact, exponential decay and can be rewritten as which equals And by some miracle* we get… This allows us to express decay in the following: Or

16. f(x) = 2x, g(x) = 0.5x

17. - Moving past basic form –Translations Graph the following: (0,4) Asymptote @ y = 3 (0,1)

18. - Moving past basic form –Translations Graph the following: (0,1) (0,0) Asymptote @ y = -1

19. - Moving past basic form –Translations We saw that: Moves our function UP “C” units And Moves our function DOWN “C” units Remember, to translate something means to "slide" an object in a fixed distance and in a given direction. The original object and its translation have the same shape and size, and they face in the same direction.

20. - Moving past basic form –Dilations Now, let’s graph the following: (0,5) (0,1) All asymptotes @ y = 0 (0,0.2)

21. - Moving past basic form –Dilations Now, let’s graph the following: (0,1) All asymptotes @ y = 0 (0,-0.2)

22. - Moving past basic form –Dilations We saw that: is identical to But if: k > 1, then stretched along the y-axis 0 < k< 1, then shrunk along the y-axis And is identical to But if: k < -1, then reflected across x-axis and stretched along the y-axis 0 < k < -1, then reflected across x-axis and shrunk along the y-axis

23. - General Form - Putting everything together from today we now have the general form of the exponential equation: Such that k, a and c are all rational numbers.

24. - Solving Exponential Equations - What does it means to solve an equation?? Oh yea…to find the roots, and how do we do that?? By setting the equation equal to 0! For this class you will be mostly solving exponential equations graphically. Throughout this topic we may occasionally come across some algebraic questions, which you have all the skills needed to solve for.

25. - Practice - Lets try: Method 1: Set equations equal and find intersection and Method 2: Set equal to 0 and find root

26. - Practice - Lets try:

27. - Practice - Lets try:

28. - Practice - Lets try:

29. - Walk Out - Simplify: 1) 3) 2) 4) Describe the properties of: Range:  Type: Asymptote: Interval: Solve the following exponential equation. First solve graphically, then attempt to solve algebraically. Show Work

30. - Walk Out - Simplify: 1) 3) 2) 4) Describe the properties of: Range: 2 > y > -inf Type: Decay Asymptote: 2 Interval: Solve the following exponential equation. First solve graphically, then attempt to solve algebraically. Show Work

31. - Warm Up - • Complete the following questions regarding the equation: • Sketch a graph • Determine the Translation • Determine the Dilation • Determine the Range • What type of Exponential • Determine the Asymptote • Solve the Equation

32. - Warm Up - • Complete the following questions regarding the equation: • Sketch a graph • Determine the Translation • 64 units down • Determine the Dilation • Stretch of 8 • Determine the Range • -64 < y < inf • What type of Exponential • Growth • Determine the Asymptote • y= -64 • Solve the Equation • (1.89,0)

33. - Applications - • When will we see exponential functions in real life? • Common Examples of Growth: • Bacteria • Compound Interest • Human Population • Common Examples of Decay: • Half-Life of radioactive isotopes • Bacteria • Animal Population • Depreciation

34. - Practice - The number of wolves in the wild, in the northern section of the Cataragas county, is decreasing at the rate of 3.5% per year.  Your environmental studies class has counted 80 wolves in the area.  After how many years will this population of 80 wolves drop below 15 wolves, if after 3 years you notice the population to be at approx. 75.83 wolves? Round to nearest year. Questions concerning growth or decay are generally written in the form of: Where: = Future Value = Initial Value a = constant base rate k = constant exponent t = time

35. - Practice - The number of wolves in the wild, in the northern section of the Cataragas county, is decreasing at the rate of 3.5% per year.  Your environmental studies class has counted 80 wolves in the area.  After how many years will this population of 80 wolves drop below 15 wolves, if after 3 years you notice the population to be at approx. 75.83 wolves? Round to nearest year. First we need to determine k Now let’s graph…

36. - Practice - The number of wolves in the wild, in the northern section of the Cataragas county, is decreasing at the rate of 3.5% per year.  Your environmental studies class has counted 80 wolves in the area.  After how many years will this population of 80 wolves drop below 15 wolves, if after 3 years you notice the population to be at approx. 75.83 wolves? Round to nearest year.

37. - Practice - The number of bacteria in a culture, N, has been modeled by the exponential function: , where t is measured in days. a) Find the initial number of bacteria in this culture. b) How many bacteria (to the nearest hundred) will there be after i) 5 days? ii) 10 days? c) How long will it take for the number of bacteria to reach 5000? d) Sketch the graph showing the number of bacteria for t≥0.

38. - Practice - The number of bacteria in a culture, N, has been modeled by the exponential function: , where t is measured in days. Find the initial number of bacteria in this culture.

39. - Practice - The number of bacteria in a culture, N, has been modeled by the exponential function: , where t is measured in days. b) How many bacteria (to the nearest hundred) will there be after i) 5 days? ii) 10 days? i) ii)

40. - Practice - The number of bacteria in a culture, N, has been modeled by the exponential function: , where t is measured in days. c) How long will it take for the number of bacteria to reach 5000? Let’s plot the graph and solve for t now!!

41. - Practice - The number of bacteria in a culture, N, has been modeled by the exponential function: , where t is measured in days. d) Sketch the graph showing the number of bacteria for t≥0.

42. - Practice - Exponential Decay of Isotopes (Half-Life) - the time required for a quantity to fall to half its value as measured at the beginning of the time period Let’s try to work through this problem: What is the half-life of Radium-226 if it’s decay constant is 0.000436? Round to nearest year. Now we graph and find the intersection…

43. - Practice - What is the half-life of Radium-226 if it’s decay constant is 0.000436. Round to nearest year. (1590,0.5)

44. - Practice - • A radioactive element decays in such a way so that the amount present each year is 95% of the amount present the previous year. At the start of 1990, there were 50 mg of the element present. • Produce a table of values showing how much of the element is present at the start of each year from 1990 to 1998. • The rule (equation) for this situation is given by N = k x at, where N mg is the amount of element present t years after the start of 1990. Find the values of a and k. • Sketch the graph N = k x at. • How long will it be before there is 25 mg of element remaining?

45. - Practice - b) The initial amount is 50 mg, therefore k = 50. The quantity is decreasing by 5% each time therefore, a = 0.95 c) d) 25 = 50 x (0.95)t Therefore t = 13.5

46. - Walk Out - The half-life of carbon-14 is known to be 5720 years. Doctor Frankenstein has 300 grams of carbon-14 in his experimental laboratory.  If untouched, how many of the 300 grams will remain after 1200 years? Round to nearest tenth and sketch a graph of your final equation. Homework pg. 307-309 (#1,3,5,8,9,11)