Chapter 5: Modeling with Exponential and Logarithmic Functions
Activity5.1: Going Shopping
SOLs: None
Objectives: Students will be able to:
Define growth factor
Determine growth factors from percent increases
Apply growth factors to problems involving percent increases
Define decay factor
Determine decay factors from percent decreases
Apply decay factors to problems involving percent decreases
Vocabulary:
Growth Factor – a percentage increase in the original value of an item
Decay Factor – a percentage decrease in the original value of an item
Key Concept:
•Growth factor problems are when the ratio of the new value to the original value is always the same and the values are increasing. This ratio is the growth factor.
•Growth factor = 1 + percentage growth
•Original value × growth factor = new value
•Decay factor problems are when the ratio of the new value to the original value is always the same, but the values are decreasing. This ratio is the decay factor.
•Decay factor = 1 - percentage decay
•Original value × decay factor = new value
Activity:
To earn revenue (income), many state and local governments require merchants to collect sales tax on the items they sell. In several localities, the sales tax is assessed at as much as 8% of the selling price and is passed on directly to the purchaser. Determine the total cost (including 8% sales tax) to the customer of the following items:
a) Greeting card selling for $1.50
b) A DVD player selling for $300
Example 1: Determine the growth factor represented by the following percent increases
a)30%
b)75%
c)15%
d)5.5%
Example 2: Determine the new values given the growth factor
a)$400 and 50% increase
b)120,000 people and a 30% increase
Example 3: Determine the growth factor for any quantity that increases 20%.
Use the growth factor to determine this year’s budget if last year’s was $75,000
Example 4: Determine the decay factor represented by the following percent decreases
a)30%
b)75%
c)15%
d)5.5%
Example 5: Determine the new values given the decay factor
a)$400 and 20% decrease
b)120,000 people and a 12% decrease
Example 6: Determine the original price of an answering machine if you got a 40% discount and paid $150.
Determine the original price of a Nordic Track if you got a 25% discount and paid $1140.
Concept Summary:
The growth factor is when things increase:
is the ratio new / original value
formed by adding specified percent increase to 100% and then changing to decimal form
original value growth factor = new value
new value growth factor = original value
The decay factor is when things decrease:
is the ratio new / original value
formed by subtracting specified percent decrease from 100% and then changing to decimal form
original value decay factor = new value
new value decay factor = original value
Homework: pg 533 – 37; problems 1 – 3, 5, 10 – 12, 15
Activity5.2: Take an Additional 20% Off
SOLs: None
Objectives: Students will be able to:
Define consecutive growth and decay factors
Determine a consecutive growth or decay factor from two or more consecutive percent change
Apply consecutive growth or decay factors from to solve problems involving percent changes
Vocabulary:
Cumulative Factors – the consecutive growth or decay factors from two or more consecutive percent changes
Key Concept: You can form a single decay (or growth) factor that represents the cumulative effect of applying the consecutive factors; the single decay factor is the product of the three decay factors. For example: a 10% off coupon on top of 25% off all Holiday items yields
(1 – 0.10)×(1 – 0.25) = (0.9)×(0.75) = (0.675)
Activity: Your friend arrives at you house. Today’s newspaper contains a 20% off coupon at Old Navy. tHe $100 jacket she had been eyeing all season was already reduced by 40%. She clipped the coupon, drove to the store, selected her jacket and walked up to the register. The cashier brought up a price of $48; your friend insisted that the price should have been only $40. The store manager arrived and re-entered the transaction, and again the registered displayed $48. Your friend left without purchasing the jacket and drove straight to your house to tell you her story.
- How do you think your friend calculated a price of $40?
- You grab a pencil and start your own calculation. First you determine the ticketed price that reflects the 40% reduction. At what price is Old Navy selling the jacket? Explain how you calculated this price.
- To what price does the 20%-off coupon apply?
- Apply the 20% discount to determine the final price of the jacket.
- If you applied the discounts in reverse order, that is, applying the 20% coupon, followed by a 40% reduction, would the final sales price change?
Example 1: A stunning $2000 gold and diamond necklace you saw was far too expensive to even consider. However, over several weeks you tracked the following successive discounts: 20% off list; 30% off marked price; and an additional 40% off every item. Determine the selling price after each of the discounts is taken.
Example 2: You purchased $1000 of a recommended stock last year and watched gleefully as it rose quickly by 30%. Unfortunately, the economy turned downward, and your stock recently fell 30% from last year’s high. Have you made or lost money on your investment?
Concept Summary:
Cumulative effect of a sequence of percent changes is the product of the associated growth or decay factors
Cumulative effect of a sequence of percent changes is the same regardless of the order the changes are applied
Homework: page542; problems 1 – 6
Activity 5.3: Inflation
SOLs: None
Objectives: Students will be able to:
Recognize an exponential function as a rule for applying a growth factor or a decay factor
Graph exponential functions from numerical data
Recognize exponential functions from equations
Graph exponential functions using technology
Vocabulary:
Exponential Function – when the independent variable appears as an exponent of the growth factor
Exponential Growth – when the independent variable appears as an exponent of the growth factor that is greater than 1
Exponential Decay – when the independent variable appears as an exponent of the decay factor that is less than 1
Key Concepts:Classic exponential functions are in the form
y = akx,
where a is any constant, and k is called the factor.
If k > 1, then it is a growth factor and
if 0 < k < 1, then k is a decay factor.
Inflation is a typical growth factor type of problem. The growth factor is 1 + inflation rate.
Depreciation is a typical decay factor type of problem. The decay factor is 1 – depreciation rate.
Activity:
Inflation means that a current dollar will buy less in the future. According to the US Consumer Price Index, the inflation rate for 2005 was 4%. This means that a one-pound loaf of white bread that cost a dollar in January 2005 cost $1.04 in January 2006. The change in price is usually expressed as an annual percentage rate, known as the inflation rate.
At the current inflation rate of 4%, how much will a $20 pair of shoes cost next year?
Assume that inflation is constant (4%) next year too; how much will the shoes cost in the year after?
Assume that inflation remains at 5% per year for the next decade. Calculate the cost of a currently priced $8 pizza for each of the next ten years and graph it.
Years From Now / Pizzas Cost0 / 8.00
1
2
3
4
5
6
7
8
9
10
Example 1: In the late 1970’s and early 1980’s inflation in the United States was a big problem. In 1980 the inflation rate was 14.3%. Gasoline was 50 cents a gallon. Assume inflation remains constant.
What is mathematical model for the cost of gas?
What is the cost of gas in 1990?
What is the cost of gas in 2010?
Example 2: You have just purchased a new car for $16,000. Much to your dismay, you have learned that you can expect the value of your car to depreciate by 15% per year (taken as soon as you drive it off the dealer’s lot).
What is the decay factor?
What is the model to represent the car’s value?
How much is the car worth after 6 years?
When will it be worth about half of its original value?
Concept Summary:
–Exponential function is a function in which the independent variable appears as an exponent of the growth factor or a decay factor
–Growth: factor greater than 1
–Decay: factor between 0 and 1
Homework: pg 547-551; problems 1, 2, 4
Activity 5.4: The Summer Job
SOLs: None
Objectives: Students will be able to:
Determine the growth or decay factor of an exponential function
Identify the properties of the graph of an exponential function defined by y = bx where b > 0 and b ≠ 1
Graph exponentials functions using transformations
Vocabulary:
Exponential Function – when the independent variable appears as an exponent of a constant
Key Concepts:
x-intercept is called a zero of the function and is the solution to a quadratic equation
Activity:
Your brother will be attending college in the fall, majoring in mathematics. On July 1, he goes to your neighbor’s house looking for summer work to help pay for college expenses. Your neighbor is interested since he needs some odd jobs done. Your brother can start right away and will work all day July 1 for 2 cents. This gets your neighbor’s attention, but you wonder if there is a catch. Your brother says that he will work July 2 for 4 cents, July 3 for 8 cents, July 4 for 16 cents and so on for every day of the month of July.
Complete the following table.
Day in July / Pay in Cents1 / 2
2 / 4
3 / 8
4 / 16
5
6
7
8
Do you notice a pattern in the output values?
Use this pattern to determine his pay on the 9th of July
Use the pattern to determine an equation that relates the day, n, to the amount of pay P(n)
What was the average rate of change between day 3 and 4?
What was the average rate of change between day 7 and 8?
Is the function linear?
Use the equation above to figure out:
What does your brother make on the 20th of July?
What would he make on the 31st of July?
Growth Factor Examples: Identify the growth factor, if any, for the given function.
a)h(x) = 1.08x
b)g(x) = 0.8x
c)f(t) = 8t
d)s(t) = 10t
Transformations: Describe in words how the graph of y = 2x is transformed by each equation.
a)y = - 2x
b)y = 5· 2x
c)y = 2x + 3 (What is the horizontal asymptote?)
d)y = 2x+3
Exponential Growth: Graph the functions f(x) = 2x and g(x) = 10x
What two things are the same for both functions?
How do they differ?
When x > 0?
When x < 0?
Decay Factor Examples: Identify the decay factor, if any, for the given function.
a)h(x) = 0.98x
b)g(x) = 1.01x
c)f(t) = 0.8t
d)s(t) = (5/8)t
Exponential Decay: Given a function g(x) = (1/2)x, answer the following:
Identify the x- and y-intercepts
Complete the following table:
x / 1 / 2 / 4 / 6 / 10g(x)
Does it have a horizontal asymptote?
What is function always doing? (Hint: slope)
Concept Summary:
Exponential functions in form y = bx, where b > 0 and b ≠ 1 have the following characteristics
–Domain is all real numbers and Range is y > 0
–Line y = 0 is a horizontal asymptote
–y-intercept is (0, 1) and there is no x-intercept
–Function is continuous
–If 0 < b < 1, then it’s a decay function; If b > 1, then it’s a growth function
Homework: pg 562 – 9; problems 4, 7, 8 - 13
Activity 5.5: Cellular Phones
SOLs: None
Objectives: Students will be able to:
Determine the growth and decay factor for an exponential function represented by a table of values or an equation
Graph exponential functions defined by y = abx, where a ≠ 0, b > 0 and b ≠ 1
Identify the meaning of a in y = abx as it relates to a practical situation
Determine the doubling and halving time
Vocabulary:
Zero-product principle – if a∙b = 0,
Key Concepts:
Exponential Functions of the form y = a∙bx, where b is > 0 and b ≠ 1
a is called the initial value, y-intercept (0, a) at x = 0
Exponential functions have successive ratios that are constant
The constant ratio is a growth factor, if y-values are increasing (b > 1)
The constant ratio is a decay factor, if y-values are decreasing ( 0 < b < 1)
Doubling time set by growth factor
Half-life is set by the decay factor
Activity:
During a meeting, you hear the familiar ring of a cell phone. Without hesitation, several of your friends reach into their jacket pockets, brief cases and purses to receive the anticipated call. Although sometimes annoying, cell phones have become part of our way of life. The following table shows the increase in the number of cell phone users in the late 1990s.
Year / Cell Phones ( in millions) / Rate of Change / Ratio between Years1996 / 44.248
1997 / 55.312
1998 / 69.14
1999 / 86.425
2000 / 108.031
Is this a linear function?
Why or why not?
Is the rate of change (slope) the same?
Is the ratio between consecutive years the same?
Does the relationship in the table represent an exponential function?
What is the growth factor?
Set up an equation, N = a∙bt, where N represents the number of cell phones in millions and t represents the number of years since 1996
What is the practical domain of the function N?
Example 1: Identify the y-intercept, growth or decay factor, and whether the function is increasing or decreasing
a)f(x) = 5(2)x
b)g(x) = ¾(0.8)x
c)h(x) = ½ (5/6)x
d)f(t) = 3(4/3)x
Example 2: An investment account’s balance, B(t), in dollars, is defined by B(t) = 5500(1.12)t, where t is the number of years.
What was the initial investment?
What is the interest rate on the account?
When will the investment double in value?
When will the investment quadruple in value?
Example 3: Chocolate chip cookie freshness decays over time due to exposure to air. If the cookie freshness is defined by f(t) = (0.8)t, find the following information.
What was the initial cookie freshness?
What is the decay rate on the cookies?
When will the cookie’s freshness be halved?
Concept Summary:
–Functions defined by y = abx, where a is the initial value and b is the growth or decay factor are exponential functions
–Y-intercept is (0, a)
–Growth factor, b > 1, y-values are increasing
–Decay factor, 0 < b < 1, y-values are decreasing
–Doubling time is the time for the y-value to double
–Half-life is the time for the y-value to be halved
Homework: pg 576 – 580; problems 2, 3, 6, 7
Activity 5.6: Population Growth
SOLs: None
Objectives: Students will be able to:
Determine annual growth or decay rate of an exponential function represented by a table of values or an equation
Graph an exponential function having equation y = a(1 + r)x
Vocabulary:
Growth rate – percentage of growth, r
Growth factor – the growth rate plus 100 percent; (1 + r)
Decay rate – percentage of decay, r
Decay factor – 100 percent minus the decay rate; (1 - r)
Key Concepts:
Linear functions represent quantities that change at a constant rate (slope)
Exponential functions represent quantities that change at a constant ratio, expressed as a percent.
Activity:
According to the 2000 US Census, the city of Charlotte, North Carolina, had a population of approximately 541,000. Assuming that the population increases at a constant rate of 3.2%, determine the population of Charlotte (in thousands) in 2001.
Determine the population of Charlotte (in thousands) in 2002.
Divide the population in 2001 by the population in 2000 and record this ratio.
Divide the population in 2002 by the population in 2001 and record this ratio
Are the ratios the same?
What does this mean?
Fill in the table below for Charlotte’s population:
Years (since 2000) / 0 / 1 / 2 / 3 / 4 / 5Population / 541
What is the growth rate?
What is the growth factor?
What is the population of Charlotte in 2006?
How long will it take for Charlotte’s population to double?
Exponential Growth Example:
Determine the growth rate, r, given the following factors:
a)b = 1.12
b)b = 1.07
c)b = 1.33
Determine the growth factor, b, given the following rates:
a)r = 5.4%
b)r = 25%
Exponential Decay Example:
Determine the decay rate, r, given the following factors:
a)b = 0.82
b)b = 0.87
c)b = 0.93
Determine the decay factor, b, given the following rates:
a)r = 6.4%
b)r = 15%
You are working at a waste-treatment facility. You are presently treating water contaminated with 18 micrograms of pollutant per liter. Your process is designed to remove 20% of the pollutant during each treatment. Your goal is to reduce the pollutant to less than 3 micrograms per liter. Complete the table:
Treatments / 0 / 1 / 2 / 3 / 4 / 5Pollutant Concentration
What percent of the pollutant remains after treatment?
What is the concentration after the first treatment?
Write the equation for the concentration, C, of the pollutant as a function on the number of treatments, n.
How many treatments are necessary to achieve the needed pollutant concentration?
Concept Summary:
–Exponential functions are used to describe phenomena that grow or decay by a constant percentage rate over time
–Annual growth rate problems are modeled by
P = P0(1 + r)t where P0 is the initial amount, r is the annual growth rate, and t is time in years
–(1 + r) represents the growth factor
–Annual decay rate problems are modeled by
P = P0(1 - r)t where P0 is the initial amount, r is the annual decay rate, and t is time in years
–(1 - r) represents the decay factor
Homework: pg 586 – 588; problems 1, 2, 5
Activity 5.7: Time is Money
SOLs: None
Objectives: Students will be able to:
Distinguish between simple and compound interest
Apply compound interest formula to determine the future value of a lump-sum investment earning compound interest
Apply the continuous compounding formula A = Pert
Vocabulary:
Compound Interest –interest earned is added to principal before the new interest is calculated
Future Value – the value of an amount in the future at a specific interest rate and compounding structure
Effective Yield – percentage by which the balance will grow in one year
Continuous Compounding – compounds interest each instant of time
Key Concepts:
Simple versus Compound Interest
Simple interest means that you earn only interest on the principal over the time period invested. Compounded interest, the interest paid on most bank’s savings accounts, pays interest on the principal and the interest it has already earned.
A = P(1 + r/n)nt
where A is the current balance
P is the principal (original deposit)
r is the annual interest rate (in decimal form)
n is the number of time per year interest is compounded
t is the time in years the money has been invested
Effective Rates
Determine the growth factor (n is number of times compounded per year)
b = ( 1 + r/n)n
Subtract 1 from b and write the result as a decimal
re = b – 1 = (1 + r/n)n – 1
Effective yield, re, will always be slightly greater than the interest rate.
Continuous Compounding
Compounded interest formula approaches A = P(1 + r/n)nt A = Pert
as the number of times the interest is compounded approaches infinity (continuous compounding). Like a horizontal asymptote on a graph. Some banks use this method for compounding interest.