CCA Ch 7: Exponential Functions
Name: ________________________________ Team: ______
7.1.1 What do exponential graphs look like? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Investigating y b x 7-1. Beginning to Investigate Exponentials Equation #1: _________________ Equation #2: _________________ Equation #3: _________________ 7-2. Investigating y b x b = ______
y
b = ______
y = ___________ x
y
y = ___________
y
x
y
x
b = ______
y
b = ______
y = ___________ x
x
y
y = ___________
y
x
y
x
b = ______
y
b = ______
y = ___________ x
x
y
y = ___________
y
x x
y x
7-3. a. What happens to y as x increases?
b. Is there an x-intercept? __________ Explain:
c. The graph has a horizontal asymptote. Does it also have a vertical asymptote? __________ Explain:
7-4. Equation: _______________________________ What is the shape of your graph?
y
x
y
What happens when x gets larger?
What happens when x gets smaller? x
How does changing the value of b change the graph?
Which aspects of the graph do not change?
Does the graph have any symmetry? If so, where?
7-5. a. Why can’t you use negative values for b?
b. How are y 1x and y 0x different from other exponential functions?
7-6. LEARNING LOG: “Investigating y b x ” Summarize what you have learned today about exponential graphs and their equations:
7.1.2 What is the connection? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Multiple Representations of Exponential Functions 7-20. a. Rebound Ratio: _______________
Work space/explanation:
Initial Height: _________________ Equation: _____________________________
b. Number of founders: ___________
Growth rate: ___________________
c. Number of computers: ___________
Portion infected each day: __________
How many will be virus-free at the end of the 3rd day? _____________
d. New equation: ____________________________________________
7-21. a. What does “a” represent?
What does “b” represent?
b. How can you identify “a” by looking at a table?
How can you find “a” in a situation?
Example of “a” in a table:
c. How can you identify “b” by looking at a table? Show how to get “b” in a table:
Example of “a” in a situation:
How can you find “b” in a situation? Show how to get “b” in a situation:
7-22. LEARNING LOG – Do this at the end of this section in “Closure”. 7-23. EQUATION → GRAPH Explain how you can create an exponential graph from the equation without making a table.
Sketch y 7(2)x according to what you explained. Be sure to mark a scale and label on each axis.
Closure: What representations have you connected so far? Give examples.
7.1.3 How does it grow? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
More Applications of Exponential Growth 7-30. SAVING FOR COLLEGE a. How does money grow with simple interest (like the example in the table)?
b. By what percent would your balance have increased from the beginning to the 4th year? Show how you know.
c. How is there $1166.40 in your account at the 2nd year?
How is compound interest calculated? How is your money growing when you receive compound interest?
d. How much money will you have in the credit-union account at the 4th year?
By what percent would your balance have increased from the beginning to the 4th year? Show how you know.
e. Which type of account grows more quickly… one with simple interest or one with compound interest? Why?
7-31. DO TOGETHER AS A CLASS – STEP GRAPHS Simple Interest Table: Year
Money Invested ($)
1800
0 1700
1 2 3
1600
4 5
1500
6 7
1400
8 Compound Interest Table: Year
1300
Money Invested ($) 1200
0 1 1100
2 3 1000
4 5 900
6 7 8
1
2
7-32. a. Simple interest equation: ____________________________ Compound interest equation: ____________________________
3
4
5
6
7
8
b. Verify that your equations in part a are correct. Enter the equations in your graphing calculator and make sure the table values match the tables in question 7-30. Adjust your equations if the tables don’t match. c.
d.
7-33. a. How much interest would you earn each quarter? __________ Set up an equation for quarterly interest: __________________________________
b. How many quarters are there in 4 years? __________ Use your equation in part a to calculate how much money you would have at 4 years:
7-34. How much money would you have if $1000 is compounded yearly at 8% for 20 years?
If you wanted to earn the same amount of money (the amount you just calculated) from a bond that earns simple interest for 20 years, what would the interest rate have to be?
Closure:
7.1.4 What if it does not grow? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Exponential Decay 7-42. THE PENNY LAB
Trial #
“Heads-Up” Pennies
0
100
a. Is it possible for a team to never remove their last penny? __________ Explain:
1 2 3
b. Would the results of this experiment have been significantly different if you had removed the “heads” each time instead of “tails”?
4 5 6
c. If you had started with 200 pennies, how would this have affected the results?
7 8
7-43. a. Before you create your graph, should your graph be continuous or discrete?
Copy your graph onto an overhead transparency. Once all teams have put their transparencies together, answer the following questions: Where does the graph cross the y-axis?
Does the graph have any asymptotes?
If so, where? b. Is this situation increasing or decreasing?
What does this mean about the value of the multiplier?
What would you expect the multiplier to be?
c. Let x = _________________________________
Let y = _________________________________
Equation: _________________________________________________
d. When x = 0, y = __________ What does that mean in the context of the problem?
e. Could there be an input for x = – 1 ? __________ If there could, what would it mean in context?
f. Domain: _______________________________ With the appropriate domain, describe what the graph would look like:
g. Where in this course have you seen graphs like the one in part a?
What is the equation for this coin situation? ______________________________________ What are the first four values of this function (using your equation)? ______ , ______ , ______ , ______ 7-44. HALF-LIFE a. If a living object is supposed to contain 100 g of carbon-14, how much would be expected to remain after 5730 years (one half-life)?
100
80
How much would remain after 11,460 years (two half-life cycles)? 60
How much would remain after three half-lives? 40
How much would remain after four half-lives?
b. Draw a graph for amount of carbon-14 (g) and # of half-lives. Label axes.
20
1
2
3
4
5
6
7
8
9
10
c. Let x = _________________________________
Let y = _________________________________
Equation: _________________________________________________
d. When x = 0, y = __________ What does that mean in the context of the problem?
e. Could there be an output for x = – 1 ? __________ If there could, what would it mean in context?
7-45. a.
12
0
3
________
0
________
5
0
________
x
0
________
0
________
0
Rule:
b.
12
c.
12
1
________
What does
12
2
________
What does
12
What does x
mean?
23
mean?
2
What does
2
1
2
mean in the context of the half-life problem?
mean in the context of the half-life problem?
7-46. a. Use your graphing calculator to compare the graphs (and tables) of y 12 What do you notice? How does a negative exponent affect its base number?
x
and y 2
x
.
b. i.
15
1
iii.
85
v.
23
vii.
1
3
32
1
ii.
100
iv.
13
vi.
6 3
1
viii.
2
2 5
Closure: What key concepts did you learn in this section?
7.1.5 What are the connections? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Graph → Equation 7-59. GRAPH → EQUATION (Assume that if there is an asymptote that it is along the x-axis.) a. Equation: ____________________________
x
y
c. Equation: ____________________________
x
y
b. Equation: ____________________________
x
y
d. Equation: ____________________________
x
y
e. Equation: ____________________________
x
y
f. Equation: ____________________________
x
y
7-60. LEARNING LOG – “Graph → Equation for Exponential Functions” Explain how to write an equation for an exponential function from its graph:
7.1.6 What is the connection? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Completing the Multiple Representations Web 7-67. WRITING A SITUATION a.
b.
c.
7-68. SITUATION → GRAPH a.
b. 2000
1600
1200
800
400
1
2
3
4
5
6
7
8
9
10
7-69. GRAPH → EQUATION a. b.
d.
e.
Equation:
___________________________________________ c.
7-70. EQUATION → GRAPH a.
7-71. SITUATION → ? a.
b.
c.
If the scale needs to be changed, indicate the scale on each axis. b.
c.
d.
7-72. LEARNING LOG – “Important Ideas about Exponential Functions” Closure: Summarize what you think is important about exponential functions:
7.2.1 How can I find the equation? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Curve Fitting and Fractional Exponents 7-79.
m = __________
b = __________
Equation: _________________________________
7-80. a. Do you agree? __________ Explain why.
b. How did she get this equation?
c. Solve for b. Show your work.
d. Equation: ______________________________________
7-81.
Equation: __________________________________ 7-82. NEW NOTATION FOR ROOTS a. Guesses about what the exponent might be:
b. What do you think?
c. Is she correct?
d. Find the value of x.
What does this tell you about another way to write “cubed root of 17”?
Check your hypothesis in your calculator.
e. Write the following roots as exponential expressions: Check your answers in your calculator.
5 _______
5
11 _______
7-83. REWRITING EXPRESSIONS-- 2 DIFFERENT WAYS FOR EACH a.
10 2/3
b.
__________ = __________
d.
( 2 )5 __________ = __________
( 3 9 )4
c.
__________ = __________
52 __________ = __________
x3
__________ = __________
3
7
e.
5
f.
y3 __________ = __________
7-84. Don’t use your calculator! 1 2
a. Re-write 27 = __________ Re-write 27
1 3
= __________
Can you get a real solution? __________ Can you get a real solution? __________
Why can you do one and not the other?
b. Re-write 27 = __________
Is it positive, negative, or no solution? ____________________
Re-write 27 = __________
Is it positive, negative, or no solution? ____________________
Re-write 27 = __________
Is it positive, negative, or no solution? ____________________
2 3
1 4
1 5
What have you discovered?
c. i. What does she mean?
ii. Is she correct?
Is she right?
7-85. Explain why y 2 is impossible to graph accurately. x
7-86. LEARNING LOG – “Zero, Negative, and Fractional Exponents” Closure: Summarize everything you know about exponents
iii. What do you think?
7.2.2 How can I find the equation? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
More Curve Fitting 7-92. a. Use Mitchell’s method to find the equation of the line through 5,15 and 3,7 .
Equation: ______________________________ b. 7-93. Use Mitchell’s method to find the equation of the line through 2,3 and 5, 6 .
Equation: ______________________________ 7-94. DISCUSS AS A CLASS Table Method for Solving the “System”: Use 2,16 and 6, 256 . Step #1: Create a table with the ordered pairs leaving space for missing values. Step #2: Calculate “b” (the multiplier). Step #3: Work backwards to calculate “a” (the starting amount).
Step #4: Write your equation: _____________________________________________
Elimination Method for Solving the “System”: Use 2,16 and 6, 256 . Step #1: Write your two equations. Set up a giant fraction with the larger exponent in the numerator. Step #2: Divide all three parts of your fraction. (The “a” values will cancel each other out.) Step #3: Use a radical to solve for “b”.
Step #4: Solve for “a” using one of the equations from Step #1.
Step #5: Write your equation: _____________________________________________ 7-95. Pick one of the two methods in the last problem to find an exponential equation for these points a.
1, 2
and 3, 162
Equation: _______________________________
b.
2, 1.75
and 2, 28
Equation: _______________________________
7.2.3 How can I use exponential functions? ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Solving a System of Exponential Functions Graphically 7-102, 103. FAST CARS Year
Concord
Escalate
Rustang
0
$27,000
$39,000
$15,000
1
$25,380
$30,225
2 3 4 5 10 n Concord:
40000
Starting Value:
38000
36000
Multiplier:
34000
32000
Equation:
30000
28000
Escalate:
26000
Starting Value: 24000
22000
Multiplier:
20000
18000
Equation:
16000
14000
Rustang:
12000
Starting Value: 10000
8000
Multiplier:
6000
4000
Equation:
2000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Which car is worth the most after one year? _________________________ Which car is worth the most after three years? _________________________ Which car is worth the most after ten years? _________________________
Which car should Jeralyn buy? _________________________
Why?
7-104. Estimate the half-life length for the Concord:
Estimate the half-life length for the Escalate:
7-105.
Equation: _____________________________________
Rate of Depreciation: _______________
EQUATION
TABLE
GRAPH
SITUATION
