8.1/8.2 Exponential Growth/Decay

1. 8.1/8.2 Exponential Growth/Decay Objectives: 1. Understand the exponential growth/decay function family. 2. Growth exponentiCl growth/decCy function using y = Cbx – h + k 3. Use exConentiCl function to models in reCl life. • Use e in the reCl world • MCniCulCte e with exConents Use e in cClculCtions with C cClculCtor GrCCh exConentiCl functions involving e VocCbulCry: Euler’s number, nCturCl bCse

2. 8.1/8.2 Exponential Growth/Decay Objectives: 1. Understand the exponential growth/decay function family. 2. Growth exponential growth/decay function using y = abx – h+ k 3. Use exponential function to models in real life. Vocabulary: Euler’s number, natural base

3. Question 1: Interest of a bank AccountYou deposit P dollars in the bank and receive an interest rate of r compounded annually for t year. Initial Asset A0 = P The end of 1st year: A1= P + P·r = P(1 + r) The end of 2nd year: A2= A1 + A1·r = A1(1 + r) = P(1 + r)(1 + r) = P(1 + r)2 The end of 3rd year: A3= A2 + A2·r = A2(1 + r) = P(1 + r)2 (1 + r) = P(1 + r)3

4. : : : t-th year: At = P(1 + r)t Since r > 0, then 1 + r > 1. Denote b = 1 + r, then b > 1. So the above model can be written as: At = P·bt (b > 1)

5. Question 2: Cell splitYou have C cells and each cell will split to 2 new cells. After n splits how many cells you will have? Initial Cell: C0 = C= C ·20 1st split: C1= C ·2 = C ·21 2nd split: C2= C1·2 = C ·22 3rd split: C3= C2·2 = C ·23 : : n-th split: Cn= C ·2n Cn = C ·bn (b = 2 > 1)

6. The observation we had from the above 2 Questions attracts us to have a deeper study to some function like: f (x) = a bx where a≠ 0, b > 0 and b ≠ 1 DefinitionExponential Function The function is of the form: f (x) = a bx, where a≠ 0, b > 0 and b ≠ 1, x  R.

7. DefinitionExponential Growth Function The function is of the form: f (x) = a bx, where a> 0, b > 1, x  R. DefinitionExponential Decay Function The function is of the form: f (x) = a bx, where a> 0, 0 < b < 1, x  R.

8. Summary a a

9. Challenge Question 1: In the definition, why b > 0 and b ≠ 1? Function Family -- functions whose graphs are identical if the graphs are shifted some units horizontally and/or vertically. ActivityUsing the graphing calculator to graph:a) y = 2 · 3xb) y = 2 · 3x+1 - 5

10. Graph b) is -- shifted 1 unit to the left of a) (horizontally)-- shifted 5 unit down of a) (vertically)-- “new” y-intercept is at (-1, -3)-- “new” asymptotes are line y = - 5 and line x = -1.New center is at (-1, - 5) Procedures1) Find the new center (h, k)2) Draw 2 perpendicular dash lines to represent new x-axis and new y-axis3) Label a + k as “new” y-intercept4) Treat horizontal and vertical dash line as “new” x-axis and “new” y-axis.

11. Example 1 Graph y = 2xy = (1/2)xy = -3 · 2xd) y = -2 (1/3)x Conclusions:1) Graph a) is an exponential growth function graph and y – intercept is 1.2) Graph b) is an exponential decay function graph and y – intercept is 1.3) Graphs c) and d) are not an exponential growth or decay function.

12. Challenge Question 2: The product of two exponential growth(or decay) functions is still an exponential growth(or decay) function? Justify your answer. Suppose that y1 = a1 b1xandy2 = a2 b2xare exponential growth function, where a1, a2 R+, b1, b2 R+, b1 > 1, b2 > 1 then since b1 b2> 1 y1y2 = a1a2 b1x b2x = a1a2(b1b2 )x is also an exponential growth function.

13. Challenge Question 3: Given an exponential growth(or decay) function y = a bx, can you construct an exponential decay(or growth) function? Given that y1 = a1 b1xis an exponential growth function, take a2 = a1 R+, b2 = 1/ b1 R+, and then 0 < b2<1 Therefore, y2 = a2 b2x is an exponential decay function.

14. Example 2 Graph y = -3 · 2x Example 3 Graph y = 3 · 4x-1 Example 4 Graph y = 4 · 3x-2 + 1

15. Example 5 Graph y = 3 · (1/2)x Example 6 Graph y = -3 · (1/4)x-1 Example 7 Graph y = 4 · (1/2)x-3 + 2

16. The variations of exponential growth/decay function model is very practical in a wide variety of application problems. Growth: y = a (1 + r ) t Decay:y = a (1 – r ) t Note: the rate r is measured the same time period as t, where a stands for the initial amount.

17. Example 8 You purchase a baseball card for $54. If it increases each year by 5%, write an exponential growth model.

18. Example 9 In 1980 wind turbines in Europegenerated about 5 gigawatt-hours of energy. Over the next 15 years, the amount of energy increased by about 5.9% per year.a) Write a model giving the amount E (gigawatt-hours) of energy and t years after 1980. About how much wind energy was generated in 1984?b) Graph the model.c) Estimate the year when 8.0 gigawatt-hours of energy were generated?

19. Example 10 You drink a beverage with 120milligrams of caffeine. Each hour, the amount c of caffeine in your system decreases by about 12%.a) Write an exponential decay model.b) How much caffeine remains in our system after 3 hours?c) After how many hours, the amount caffeine in your system is 50 milligrams of caffeine?

20. Example 11 You have a new computer for $2100.The value of the computer decreases by about 30% annually.a) Write an exponential decay model for the value of the computer. Use the model to estimate the value after 2 years.b) Graph the model.c) Estimate when the computer will be worth $500?

21. 8.1/8.2 Exponential Growth/Decay Assignment: 8.1 P469 #14-40 even 8.1 P469 #43-48, 56, 57, 66 #16-47