Lesson 7.1.2
HW: 7-24 to 7-29
Learning Target: Scholars will deepen and extend their understanding of exponential functions by
examining the multiplier (“b”) and starting point (“a”) in different representations. Students will
generalize the roles of a and b for the equation 𝑦 = 𝑎(𝑏)𝑥 .
In Chapter 2 you looked at multiple representations (such as a table, graph, equation, or situation) of
linear functions. In this chapter you will use multiple representations to learn more about the multiplier
and starting point of exponential functions.
7-20. Let’s look at some of the connections between the multiple representations of an exponential
function.
1.
Arnold dropped a ball during the bouncing ball activity and recorded
its height in a table. Part of his table is shown at right. What was the rebound ratio of his
ball? At what height did he drop the ball? Write an equation that represents his data.
Explain your equation.
2. A major technology company, ExpoGrow, is growing incredibly fast. The latest
prospectus (a report on the company) said that so far, the number of employees, y, could
be found with the equation y = 3(4)x , where x represents the number of years since the
company was founded. How many people founded the company? How can the growth of
this company be described?
3.
A computer virus is affecting the technology center in such a
way that each day, a certain portion of virus-free computers is infected. The number of
virus-free computers is recorded in the table at right. How many computers are in the
technology center? What portion of virus-free computers is infected each day? How
many computers will remain virus-free at the end of the third day? Justify your answer.
4. As part of a major scandal, it was discovered that several statements in the prospectus for
ExpoGrow in part (b) were false. If the company actually had five founders and doubles
in size each year, what equation should it have printed in its report?
7-21. Most of the exponential equations you have used in this chapter have been in the form y = abx .
Explore using the Exponential Functions Student eTool (Desmos).
5. What does a represent in this equation? What does b represent?
6. How can you identify a by looking at a table? How can you find it in a situation? Give an
example for each representation.
7. How can you determine b in each representation? Use arrows or colors to add your ideas
about b to the examples you created in part (b).
7-22. MULTIPLE-REPRESENTATIONS WEB
What connections are you sure you can use in an exponential functions web? For example, if you have an
exponential equation, such as y = 20(3)x , can you complete a table If so, draw an arrow from the
equation and point at the table, as shown at right.
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Copy the web, without any arrows, into your Learning Log. Discuss with your team the
connections you have used so far in this chapter. Draw arrows to show which representations you
can connect already. Which connections have you not used yet but you are confident that you
could? Which connections do you still need to explore?
Can you think of examples from earlier today, or in Chapter 5, to support your conclusions?
Write down the problem numbers next to your arrows.
Title this entry “Multiple-Representations Web for Exponential Functions” and label it with
today’s date. Be ready to share your findings with the rest of the class.
7-23. EQUATION → GRAPH
How can you sketch the graph of an exponential function directly from its equation without making a
table first? Discuss this with your team. Then make a reasonable sketch of the graph of y = 7(2)x on
your paper.
7-24. Each table below represents an exponential function of the form y = abx . Copy and complete each
table on your paper and find the corresponding equation.
1.
2.
7-25. Brianna is working on her homework. Her assignment is to come up with four representations for an
exponential function of her choosing. She decides it is easiest to start by writing an equation, so she
chooses
(Desmos).
. Help Brianna create the other three components of the web. 7-25 HW eTool
7-26. Sketch the graphs of y = x2, y = 2x2, and
on the same set of axes. Describe the similarities
and differences among the graphs. 7-26 HW eTool (Desmos).
7-27. Write an equation or system of equations to solve this problem.
Morgan started the year with $615 in the bank and is saving $25 per week. Kendall started with $975 and
is spending $15 per week. When will they both have the same amount of money in the bank?
7-28. Examine each sequence below. State whether it is arithmetic, geometric, or neither. For the
sequences that are arithmetic, find the formula for t(n). For the sequences that are geometric, find the
sequence generator for t(n).
3.
4.
5.
6.
7.
8.
1, 4, 7, 10, 13, …
0, 5, 12, 21, 32, …
2, 4, 8, 16, 32, …
5, 12, 19, 26, …
x, x + 1, x + 2, x + 3 …
3, 12, 48, 192, …
7-29. Scientists hypothesized that dietary fiber would impact the blood cholesterol level of college
students. They collected data and found r = –0.45 with a scattered residual plot. Interpret the scientists’
findings in context.
Lesson 7.1.2
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7-20. See below:
1. y = 132(0.8)x, rebound ration 0.8, initial height ≈ 132 cm
2. This company had three founders and quadruples in size each year.
3. The technology center has 27 computers, and
of the virus-free computers are
infected each day, so 8 computers will be virus-free on the 3rd day.
4. y = 5(2)x
7-21. See below:
1. a represents the initial value, and b represents the multiplier.
2. The value of a can be found when x = 0; so in a table, look for the y-value when x = 0; on
a graph, locate the y-intercept; and in a situation, a is often the starting value (such as
when no time has passed).
3. On a graph, you can divide the y-value for any point by the y-value for the point with xvalue one unit lower; in a table, you can divide the value of y when x = 1 by the value of
y when x = 0; and in a situation, the b-value is the multiplier.
7-22. Answers vary, but at this point, it is expected that students will feel confident about the
following connections: situation ↔ equation, equation ↔ table, table ↔ situation, and table
↔ graph. Students are probably not very comfortable going directly from equation ↔ graph,
or situation ↔ graph, without making a table first.
7-23. See graph below:
7-24. See below:
1. See table below. y = 1.2(3.3)x
2. See table below. y = 5·6x
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7-25. Answers vary but should include a table, a graph, and a situation.
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7-26. They are all parabolas, with y = 2x2 rising most rapidly and y =
solution graph below.
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7-27. 9 weeks
7-28. See below:
1. arithmetic, t(n) = 3n – 2
2. neither
3. geometric, r = 2
4. arithmetic, t(n) = 7n – 2
5. arithmetic, t(n) = n + (x – 1)
6. geometric, r = 4
7-29. There is a weak negative linear association: as dietary fiber is increased, blood cholesterol
drops. 20.25% of the variability in blood cholesterol can be explained by a linear association with
dietary fiber.
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x2 most slowly. See