Exponential Function
(Day 3 & 4)How do transformations of exponential equations affect the function analytically and graphically? (Day 5)How do exponential functions relate to real world phenomena? | |
b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrea c. Graph functions as transformations of f(x) = ax. d. Solve simple exponential equations and inequalities analytically, graphically, and by using | |
Opening – |
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Work session – |
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(Day 4) Have the students share their answers from the work session. (Day 5)Have the students share their answers and how it relates to the standards. |
Exponential Function
y = ax
1. Graph f(x) = 3x.
Complete the table of values. | |
x | f (x) |
– 4 | |
– 2 | |
0 | |
1 | |
3 | |
4 | |
2. Graph f(x) = 5x.
Complete the table of values. | |
x | f (x) |
– 2 | |
0 | |
1 | |
3 | |
4 | |
3. Graph f(x) = 3x + 2.
x | f (x) |
– 4 | |
– 2 | |
0 | |
1 | |
3 | |
4 | |
How did adding the 2 change
the original function?
the original function?
4. Graph f(x) = 5x – 3.
Complete the table of values. | |
x | f (x) |
– 4 | |
– 2 | |
0 | |
1 | |
3 | |
4 | |
How did subtracting the 3 change
the original function?
the original function?
Exponential Function GO 2
Exponential Function
y = ax
f (x) = 5x + 1 | f (x) = – 5x | ||
x | f (x) = | x | f (x) = |
– 2 | – 2 | ||
– 1 | – 1 | ||
0 | 0 | ||
1 | 1 | ||
2 | 2 | ||
3 | 3 | ||
f (x) =(0.125)3x | f (x) = (2)5x | |||
x | f (x) = | x | f (x) = | |
– 2 | – 2 | |||
– 1 | – 1 | |||
0 | 0 | |||
1 | 1 | |||
2 | 2 | |||
3 | 3 | |||
f (x) = 5x – 3 | f (x) = 3x + 2 | ||
x | f (x) = | X | f (x) = |
– 2 | – 2 | ||
– 1 | – 1 | ||
0 | 0 | ||
1 | 1 | ||
2 | 2 | ||
3 | 3 | ||
Exponential Function GO 3
Function: f(x) = ( – 2 ) 3x + 2
Domain______________________
Range________________________
Asymptotes____________________
Zeros________________________
y-intercepts___________________
intervals of decrease______________
intervals of increase_______________
rates of change___________________
Function: f(x) = (0.85) 3x – 2 – 1
Domain______________________
Range________________________
Asymptotes____________________
Zeros________________________
y-intercepts___________________
intervals of decrease_______________
intervals of increase_______________
rates of change___________________
Function: f(x) = (2) 3x – 1 + 2
Domain______________________
Range________________________
Asymptotes____________________
Zeros________________________
y-intercepts___________________
intervals of decrease_______________
intervals of increase_______________
rates of change___________________
Unit 5 Exponential Functions
Day 5 Quiz
Simplify using the exponential properties.
1. (x 3 y2) (x3 y7)=
2. X3y
X2y
3. Solve the exponential equation-
(a) 3x+1 = 27x+3 (b) 9x+2 = (1/27)x+12
Solve each inequality.
4. 8x ≥ 22x+1
5. 4x ≤ 43x-1
6. #23 p. 130 word problems (McDougal Littell – possibly)
Exponential Decay Experiments
The following are experiments to explore the phenomena of exponential decay. Each of these experiments will require you to collect materials, take repeated measurements and graph the resulting data.
Experiment 1: Cooling Water
You will need: a container of hot water, a watch, and a candy thermometer
1. For this experiment, you need to measure the temperature of the hot water. Record this measurement in the table below. Recheck and record the temperature of the water every minute until the water reaches room temperature.
x (time in minutes) | y (temperature of water) |
0 | |
1 | |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 | |
16 |
2. Graph the ordered pairs (time, temperature) on graph paper.
3. Does this data appear exponential? Why or why not?
4. Use the initial value you recorded plus one other point from your graph to write an exponential function to fit your curve.
points you chose: _________________________
f(x) = _________________________________
5. Using your equation, f(x), complete the following table. Plot the new values (x, f(x)) on your graph in another color.
x (time) | f(x) |
0 | |
1 | |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 | |
16 |
6. Sketch your curve f(x). Does it appear to be a good fit?
7. What is the “decay factor”?
Part 3: The Beginning of a Business
How in the world did Linda ever save enough to buy the franchise to an ice cream store? Her mom used to say, “That Linda, why she could squeeze a quarter out of a nickel!” The truth is that Linda learned early that patience with money is a good thing. When she was just about 9 years old, she asked her dad if she could put her money in the bank. He took her to the bank and she opened her very first savings account.
Each year until Linda was 16, she deposited her birthday money into her savings account. Her grandparents (both sets) and her parents each gave her money for her birthday that was equal to her age; so on her ninth birthday, she deposited $27 ($9 from each couple).
Linda’s bank paid her 3% interest, compounded quarterly. The bank calculated her interest using the following formula.
where A = final amount, P = principal amount, r = interest rate, n = number of times per year the interest was compounded
1. Using the following chart, calculate how much money Linda had on her 16th birthday.
Age | Birthday $ | Amt from previous year plus Birthday | Total at year end |
9 | 27 | 0 | 27.81916 |
10 | 30 | 57.81916 | 90.48352 |
11 | 33 | 123.4835 | 161.2311 |
12 | 36 | 197.2311 | 240.3071 |
13 | 39 | 279.3071 | 327.9643 |
14 | 42 | 369.9643 | 424.463 |
15 | 45 | 469.463 | 530.0714 |
2. On her 16th birthday, the budding entrepreneur asked her parents if she could invest in the stock market. She studied the newspaper, talked to her economics teacher, researched a few companies and finally settled on the stock she wanted. She invested all of her money in the stock and promptly forgot about it. When she graduated from college on her 22nd birthday, she received a statement from her stocks and realized that her stock had appreciated an average of 10% per year. How much was her stock worth on her 22nd birthday?
3. When Linda graduated from college and got her first job, she decided that each year on her birthday she would purchase new stock in the amount of half what her last stock was worth. On her 30th birthday she looked back and saw that her stock had appreciated each year a percent that was half of her age that year. So on her 23rd birthday, her stock had appreciated 11.5%; and so on. What was her stock worth on her 30th birthday?
Age | Amt from previous year | Amt Linda added | Amt at year end |
22 | 938.73 | 469.47 | |
23 | |||
24 | |||
25 | |||
26 | |||
27 | |||
28 | |||
29 | |||
30 | 147,888.83 |
Part 4: Some Important Questions
All of these examples from Linda’s journey are examples of exponential growth functions… the rumor, compounding interest in a savings account, appreciation of a stock. Real-life situations tend to have restricted domains.
1. How is the domain restricted in each of the scenarios?
2. How would the graph of the rumor be different if the domain was unrestricted?
3. Graph the function
4. What is the range of the function?
5. Why doesn’t the graph drop below the x-axis?
6. Now graph
7. What is the range of the function?
8. An exponential function has a horizontal asymptote. Where is the asymptote located in the graph for #3? Where is the asymptote located in the graph for #6?
9. Use your graphing utility to graph the following equations.
Ø
Ø
Ø
Ø
Ø
10. Make some generalizations. What impact did each of the changes you made to the equation have on the graph?
shifts how?
Day 6 | |
E. Q. – | How do exponential functions relate to real world phenomena? |
Standard – | MM2A2. Students will explore exponential functions. e. Understand and use basic exponential functions as models of real phenomena. |
Opening – |
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Work session – | Students will work two problem McDougal Littell Mathematics II textbook.(Compound interests problems) 1. Pg 125 mathII example 4 2. Pg 127 mathII #18-20 3. Students will make a problem, solve it, graph it & discuss what occurred. (in-class & homework) |
Closing – | Students share out work and answer EQ if they can. |
Mathematics II - Unit 5
Real World Phenomena Worksheet
INVESTMENTS
Consider a $1000 investment that is compounded annually at three different interest rates:
5%, 5.5%, and 6%.
a. Write and graph a function for each interest rate over a time period from 0 to 60 years.
b. Compare the graphs of the three functions.
c. Compare the shapes of the graphs for the first 10 years with the shapes of the graphs between 50 and 60 years.
24. Which of these describes the graph of f(x) = 3x + 4?
A. It has a vertical asymptote at x = 0.
B. It has a vertical asymptote at x = -4.
C. It has a horizontal asymptote at y = 0.
D. It has a horizontal asymptote at y = 4.
25. What is the asymptote of the graph of f(x) = 2x?
A. x-axis
B. y-axis
C. y = 1
D. y = -1
26. How would you translate the graph of f(x) = 5x to produce the graph of
f(x) = 5x – 3?
- translate the graph of f(x) = 5x left 3 units
- translate the graph of f(x) = 5x right 3 units
- translate the graph of f(x) = 5x up 3 units
- translate the graph of f(x) = 5x down 3 units
29. Find the y-intercept of the graph of y = -3(7x) .
2. D 23. A
3. D 24. D
4. A 25. A
5. C 26. D
6. D 27. C
7. D 28. B
8. B 29. C
9. A 30. B
10. D 31. D
11. A
12. D
13. D
14. A
15. A
16. B
17. D
18. B
19. C
20. A
KNOW | HOW | NOW |
Function | One-to-Oneness | |
Domain | Inverse | |
Range | Inverse Relation | |
Intercepts | Inverse Function | |
Maximum | Composition | |
Minimum | Composite | |
Linear Function | Horizontal Line Test | |
Quadratic Function | Power Function | |
Cubic Function | Restricted Domains | |
Vertical Line Test | nth Root |
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