Integrated Math I Name: Period: Date: Sec 8.2.3 Solving a ...

Integrated Math I

Name: ________________________________________ Period: ______ Date: _______________

Sec 8.2.3 Solving a System of Exponential Functions Graphically

Do Now

1. A multiplier is the number ___________________ times each term to get the next term.

2. The general equation of an exponential function is _____________, where a is the ______________________________ and b is the _____________________________.

3. To "undo" a power of 3 (x3) you _____________________________ ! . That is, ! !.

4. To "undo" a power of 4 (x4) you _____________________________ ! . That is, ! !.

5. Solve 1,400 = 1,000b4 for b.

6. If the multiplier is 0.94, then the _________ rate is ____________________________.

7. To sketch graphs of exponential functions--you have growth when b is _______________ and decay when b is _______________.

Problems Highlight important information and circle the prompt(s)/ question(s)

In this lesson, you will apply your skills with exponential functions to a system of equations as you explore the value of cars in an investigation.

8-115. FAST CARS. The moment you drive a new car off the dealer's lot, the car is worth less than what you paid for it. This phenomenon is called depreciation, which means that as time goes on the car is worth less and less. Some cars depreciate more than others (that is, they depreciate at different rates), but most cars depreciate over time. On the other hand, some older cars actually increase in value over time. This is called appreciation. Jeralyn had a choice between buying a new Concord EX for $27,000, a used Escalate that would have been $70,000 new if bought last year, or a 1967 Tustang for $18,150. Consumers Car Resource reports that the value of the Concord depreciates at 6% per year and the Escalate will be worth $28,672 three years from now. It also reports that the appreciation of a Tustang has historically been very consistent, and it expects the trend to continue:

Historical Values for 1967 Tustang

Two Years Ago

$15,000

Last Year

$16,500

This Year

$18,150

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Integrated Math I

Name: ________________________________________ Period: ______ Date: _______________

8-116. Investigate the changing values of each of the cars by addressing the questions below.

a. What is the multiplier for the Concord? For the Escalate? For the Tustang? Justify your answer.

Concord

Escalate

Tustang

b. Make a table like the one below and calculate the value for each car for each year shown.

c. Graph the data for all three cars on the same set of axes. Are the graphs linear? How are they similar? How are they different? You may want to use a different color for each car. Response:

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Integrated Math I

Name: ________________________________________ Period: ______ Date: _______________

d. Write a function to represent the value of each car.

Concord

Escalate

Tustang

e. How much will the value (in dollars) of each car change during the two years from today? Justify your

answer.

Concord

Escalate

Tustang

f. Using the graph, which of the three cars is worth the most after one year? After five years? In how many years will the values of the Concord and Tustang be the same?

After one year, the Escalate will be worth the most. After five years, the Tustang will be worth the most. The Concord and Tustang will have the same value after about 2.5 years.

g. Pick one of the three cars and explain why Jeralyn should buy it. Has this problem changed your view of buying cars?

Answers vary.

8-118. In 2011, a brand new SUV cost $35,000 to drive off the lot. In 2014, that same SUV was valued at $22,500. Write an exponential equation to represent this information. What is the rate of depreciation for the SUV?

The equation is 22,500 = 35,000b3, where b is the multiplier. To find the multiplier, solve for b in the equation 22,500 = 35,000b3 ? 0.642 b3 ? 0.863 b. Thus the SUV is deprecating at a rate of 13.7% every year because 1 ? 0.863 = 0.137 ?13.7

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Integrated Math I

Name: ________________________________________ Period: ______ Date: _______________

8-119. Over a four-month period the price of an ounce of gold increased exponentially from $1000 to

$1400. What was the monthly multiplier? What was the monthly percent increase? Write an equation that

models this situation.

The information given can be model by the equation 1,400 = 1,000b4, where b is the multiplier. To find the multiplier, solve for b in the equation 1,400 = 1,000b4? 1.4 b4 ? 1.088 b. The monthly increase is 8.8% since 1 ? 1.088 = 0.088 ?8.8. This situation is modeled by the equation y = 1000(1.088)x, where x is the number of months and y is the price of an ounce of gold.

8-121. An exponential function of the form f(x) = abx that passes through the points (2, 2) and (4, ?).

a. What is the equation of the function?

b. Use the equation to sketch a graph of the function.

8-123. If f(x) = 3(2)x, what are the value of the expressions in parts (a) through (c) below? Then complete parts (d) through (f).

a. f(-1) = 3(2)-1= 3/2

b. f(0) = 3(2)0 =3

c. f(1) = 3(2)1= 6

d. What value of x gives f(x) = 12?

12 = 3(2)x ? 4 = 2x ? x = 2

Homework Study for chapter test and make flashcard 4

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