Section 1 - Quia



Activity 5.1: 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

Activity 5.2: 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.

1. How do you think your friend calculated a price of $40?

2. 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.

3. To what price does the 20%-off coupon apply?

4. Apply the 20% discount to determine the final price of the jacket.

5. 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: page 542; 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 Cost |

|0 |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 Cents |

|1 |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 |10 |

|g(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 Years |

|1996 |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 |5 |

|Population |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 |5 |

|Pollutant 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.

Activity:

Congratulations, you have inherited $20,000! Your grandparents suggest that you save half of the inheritance for a “rainy day.” Suppose the $10,000 is deposited in a bank at 6.5% annual simple interest. What is the interest earned after 1 year?

What would be the interest in 10 years?

Simple vs Compound Interest Example:

Suppose you won (tax-free) a million dollars and deposited it in an account earning 5% interest simple interest. How much will you have after 10 years?

If it was deposited in an account earning 5% compounded yearly, how much will you have after 10 years?

Compounding Comparisons:

Determine the effective yield associated with each of the growth factors (interest rate of 6.5% with different compounding schedules) in the following table.

|n |1 |4 |12 |365 |

|Growth Factor, b |1.065 |1.0666 |1.06697 |1.06715 |

|Effective Yield, re | | | | |

|How Compounded | | | | |

Does the number of times compounded make a difference?

Effective Yield Example:

Determine the effective yields for the following rates and compounding schedules:

a) r = 4.5%, compounded monthly

b) r = 2.5%, compounded quarterly

c) r = 5.5%, compounded daily

Compounding Example:

Calculate the balance of your $10,000 investment in 10 years with an annual interest rate of 6.5% compounded continuously.

What is the growth factor in this case?

What is the effective yield?

Retirement Example:

Historically, investments in the stock market have yielded an average rate of 11.7% per year (over the long haul). Suppose on graduating high school a rich aunt deposits 10,000 in an account at an 11% annual interest rate that compounds continuously for your retirement. She claims that you will have over a million dollars by retirement time (age 65). Is she right?

Concept Summary:

– Compound interest formula: A = P(1 + r/n)nt

where A = current balance

P is the principal (original deposit)

r is the annual interest rate (in decimal form)

n is the number of times per year of compounding

t is the time in years money is invested

– Continuous compounding formula: A = Pert

– If the number of compounding periods is large, then the compound interest formula can be approximated by continuous compounding formula.

Homework: pg 595 – 597; problems 1-3

Activity 5.8: Continuous Growth and Decay

SOLs: None

Objectives: Students will be able to:

Discover the relationship between the equations of exponential functions defined by y = abt and the equations of continuous growth and decay exponential functions defined by y = aekt

Solve problems involving continuous growth and decay models

Graph base e exponential functions using transformations

Vocabulary:

Vary Directly – if

Key Concept:

Continuous growth is modeled by the equation: y = aekt

where a is the initial amount, k is the constant continuous growth rate and t is time

Activity:

The US Census Bureau reported that the US population on April 1, 2000 was 281,421,906. The US population on April 1, 2001 was 284,236,125. Assuming exponential growth, the US population y can be modeled by the equation y = abt, where t is the number of years since April 1, 2000 (when t = 0).

What is the initial value, a?

What is the annual growth factor, b?

Assuming exponential growth, the US population y can be modeled by the equation y = abt.

What is the annual growth rate?

What is the equation for US population as a function of t?

Use this to estimate the US population on 1 Apr 2011.

Change the equation y = abt, to a continuous growth form of y = aekt. So bt = ekt and from algebra ekt = (ek)t

How are b and ek related?

Using our calculator, let Y1 = ex and Y2 = 1.01 and find their intersection (solution for b = ek)?

Rewrite the US population function in continuous growth format.

Example 1: A bacterial growth in a culture increases by 25% every hour. If 10000 are present when the experiment starts:

Determine the constant, k, in continuous growth model

Write the equation for the continuous model

When will the sample double?

Example 2: Tylenol (acetaminophen) is metabolized in your body and eliminated at a rate of 24% per hour. You take two Tylenol tablets (1000 milligrams) at 1200 noon.

What is the initial value?

Determine the decay factor, b.

Find the constant continuous decay rate, k.

Write the continuous decay function

Example 3: Compared to y = ex, describe the graphic relationship between its graph and the following graphs:

a) y = - ex

b) y = ex+2

c) y = ex + 2

d) y = 2ex

e) y = e-x

f) y = 1 – 2ex

Concept Summary:

– Quantities that increase or decrease continuously at a constant rate can be modeled by y = aekt.

– Increasing: k > 0 k is continuous rate of increase

– Decreasing: k < 0 |k| is continuous rate of decrease

– The initial quantity at t=0, a, may be written in other forms such as y0, P0, etc

– Remember the general shapes of the graphs

Homework: pg 604-09; problems 2, 3, 8

Activity 5.9: Bird Flu

SOLs: None

Objectives: Students will be able to:

Determine the equation of an exponential function that best fits the given data

Make predictions using an exponential regression equation

Determine whether a linear or exponential model best fits the data

Vocabulary: None new

Activity:

In 2005, the avian flu, also known as the bird flu, received international attention. Although there were very few documented case of the avian flu infecting humans worldwide, world health organizations including the Centers for Disease Control in Atlanta expressed concern that a mutant strain of the bird flu virus capable of infecting humans would develop and produce a worldwide pandemic. The infection rate (the number of people that any single infected person will infect) and the incubation period (the time between exposure and the development of symptoms) of this flu cannot be known precisely but they can be approximated by studying the infection rates and incubation periods of existing strains of the virus.

A conservative infection rate would be 1.5 and a reasonable incubation period would be about 15 days (0.5 months). This means that the first infected person could be expected to infect 1.5 people in about 15 days. After 15 days that person cannot infect anyone else. This assumes that the spread of the virus is not checked by inoculation or vaccination. So after 15 days from the first infection there are 2.5 people infected. In the next 15 days the 1.5 newly infected people would infect 2.25 more people (1.5 ( 1.5). Now we have 4.75 people who have been infected (at the end of one month).

Fill in the following table (Round each value to the nearest person):

|Months |0 |0.5 |1 |

|Golf Ball |0.043 |Moon |3,476,000 |

|Baseball |0.075 |Earth |12,756,000 |

|Basketball |0.239 |Jupiter |142,984,000 |

If we wanted to plot diameter as the x-variable and either volume or surface area as the y-variable, then we would need to put all these values on the horizontal axis somehow.

If we plot the first three on the axis below

[pic]

If we plot the last three on the axis below

[pic]

If we scale the axis in exponential powers of 10 (each tick mark is ten times larger than the one before), then we can fit them all on the same scale. This is called a logarithmic scale.

[pic]

Many times a logarithmic scale will be marked with just the exponents (and a note about it being logarithmic).

[pic]

Richter scale (measuring earthquakes)

The magnitude value is proportional to the logarithm of the amplitude of the strongest wave during an earthquake. A recording of 7, for example, indicates a disturbance with ground motion 10 times as large as a recording of 6. The energy released by an earthquake increases by a factor of 30 for every unit increase in the Richter scale. The table below gives the frequency of earthquakes and the effects of the earthquakes based on this scale

|Richter scale |No. of earthquakes per year |Typical effects of this magnitude |

|< 3.4 |800 000 |Detected only by seismometers |

|3.5 - 4.2 |30 000 |Just about noticeable indoors |

|4.3 - 4.8 |4 800 |Most people notice them, windows rattle. |

|4.9 - 5.4 |1400 |Everyone notices them, dishes may break, open doors swing. |

|5.5 - 6.1 |500 |Slight damage to buildings, plaster cracks, bricks fall. |

|6.2  6.9 |100 |Much damage to buildings: chimneys fall, houses move on foundations. |

|7.0 - 7.3 |15 |Serious damage: bridges twist, walls fracture, buildings may collapse. |

|7.4 - 7.9 |4 |Great damage, most buildings collapse. |

|> 8.0 |One every 5 to 10 years |Total damage, surface waves seen, objects thrown in the air. |

Example 1: Evaluate each of the following

a) Log 104

b) Log (1/100)

c) Log 1000

d) Log4 64

e) Ln e

f) Log3 (1/27)

Example 2: Evaluate each of the following with your calculator

a) Log 20

b) Ln 15

c) Log 0.02

Rewrite each logarithmic expression as exponential:

a) 3 = log2 8

b) 0 = log5 1

Concept Summary:

– Notation of logarithms is y = logb x

where b is the base of the log, x is the resulting power of b and y is the exponent

– Common logs (base 10) y = log x

– Natural logs (base e) y = ln x

– y = logb x is equivalent to exponential by = x

– Special log properties:

– logb 1 = 0

– logb b = 1

– logb bn = n

Homework: pg 632 – 633; problems 1- 5

Activity 5.11: Walking Speed of Pedestrians

SOLs: None

Objectives: Students will be able to:

Determine the inverse of the exponential function

Identify the properties of the graph of a logarithmic function

Graph the natural logarithmic function using transformations

Vocabulary:

Logarithmic function – defined by the inverse function of an exponential function to the base, b

Inverse functions – interchange the domain and range of the original function. The graph of an inverse function is the reflection of the original function about the line y = x. You determine the inverse function by switching x and y and solve the new equation for y.

Vertical asymptote – a vertical line that the function approaches but never reaches (usually a restricted value in the domain)

Key Concept:

The logarithmic function base b is defined by y = logb x

where b is the base (b > 0 and b ≠ 1)

x is the power of the base, b and y is the exponent needed on b to obtain x

Properties of f(x) = log x (common log)

• Base is 10

• Domain of x is set of all positive numbers (x > 0)

• Range of y is set of all real numbers

• f(x) is inverse of g(x) = 10x

Properties of f(x) = ln x (natural log)

• Base is e (natural number ≈ 2.718281828)

• Domain of x is set of all positive numbers (x > 0)

• Range of y is set of all real numbers

• f(x) is inverse of g(x) = ex

Activity: On a recent visit to Boston, you notice that people seem rushed as the move about the city. Upon returning to school, you mention this observation to your psychology teacher. The instructor refers you to a psychology study that investigates the relationship between the average walking speed of pedestrians and the population of the city. The study cites statistics presented in the graph.

[pic]

Does the data appear to be linear?

Does the data appear to be exponential?

Natural Logarithm

Fill in the table and graph y = ln x

|x |0.01 |0.1 |0.25 |0.5 |

|-2 | | |0.01 | |

|-1 | | |0.1 | |

|0 | | |1 | |

|1 | | |10 | |

|2 | | |100 | |

Find the inverse of the following functions

a) y = 2x – 3

b) y = x2

c) y = 5x

Natural Logarithmic Transformations

Compared to y = ln x, describe the graphic relationship between its graph and the following graphs:

a) y = -ln x

b) y = ln (x + 3)

c) y = ln (-x)

d) y = ln x + 3

e) y = 2 ln x

f) y = 3 – 2 ln x

Concept Summary:

– Properties of the function y = log x:

– Domain is x > 0

– Range is all real numbers

– Is the inverse of the function x = 10y

– Graph of the function y = logb x or y = ln x:

– is increasing for all x > 0

– x-intercept of (1,0) and no y-intercept

– x=0 (y-axis) is a vertical asymptote

– is continuous for x > 0

– Inverse functions are determined by interchanging x and y variables

Homework: pg 640-43; problems 1, 3-5

Activity 5.12: Walking Speed of Pedestrians, continued

SOLs: None

Objectives: Students will be able to:

Compare the average rate of change of increasing logarithmic, linear, and exponential functions

Determine the regression equation of a natural logarithmic function, y = a + b ln x, that best fits a set of data

Vocabulary: None new

Key Concept:

Let’s compare three of the major function families that we have studied this year:

linear, exponential, and logarithmic.

Graph the following: Y1 = x; Y2 = ex; and Y3 = ln x

And compare them using the following table:

|Function |Slope |Grows |Horizontal Asymptote |Vertical Asymptote |Domain |Range |

|y = x | | | | | | |

|y = ex | | | | | | |

|y = ln x | | | | | | |

This illustrates the relationship between f(x) = bx, b > 1; g(x) = logb x, b > 1 and y = mx + b, m > 0

Activity: In Activity 5-11 we looked at a psychological study that investigated the relationship between the average walking speed of pedestrians and the population of the city. The table below summarizes the data:

|Population (in K) |50 |100 |200 |500 |1000 |2000 |

|Ave Walking Speed |3.7 |4.3 |4.9 |5.7 |6.3 |6.9 |

Run a natural logarithmic regression on the data

What is the model? y = a + b ln x

What is the practical domain of the function?

Predict the walking speed in Boston (pop = 589,121)

Predict the walking speed in New York (pop = 8,008,278)

If the average walking speed is 5.2, write an equation that can be used to find the population, P, of the city.

How do we solve it graphically?

How do we solve it algebraically?

Concept Summary:

– As the input of a logarithmic function increases, the output increases at a slower rate (the slope is smaller; the graph becomes less steep)

– The relationships between f(x) = bx, g(x) = logb x and y = mx + b are seen in the table below

|Function |Direction |Growth Rate |Asymptote |Domain |Range |

|f(x) = bx b > 1 |Increasing |Fastest |Horizontal |All real |Y > 0 |

|g(x) =logbx b > 1 |Increasing |Slowest |Vertical |X > 0 |All real |

|y = mx + b m > 0 |increasing |constant |none |All real |All real |

Homework: pg 649 – 654; problems 2, 6

Activity 5.13: The Elastic Ball

SOLs: None

Objectives: Students will be able to:

Apply the log of a product property

Apply the log of a quotient property

Apply the log of a power property

Discover change of base formula

Vocabulary: None new

Key Concept:

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Activity: You are continuing your work on the development of the elastic ball. You are still investigating the question, “If the ball is launched straight up, how far has it traveled vertically when it hits the ground for the 10th time?” However, your supervisor tells you that you cannot count the initial launch distance. You must calculate only the rebound distance.

Using some physical properties, timers and your calculator, you collect the following data.

Plot the data using STATPLOT

What does the graph say about a model: linear, exponential or logarithmic?

The data can be modeled (thru logarithmic regression) as T = 26.75 log N

Plot both and see if the model is reasonable

Product Examples:

Log2 32 =

Log (5st) =

Log4 3 + log4 9 =

Log2 7 + log2 11 =

Quotient Examples:

Log2 (2x/y) =

Log (¾st) =

Log4 3 - log4 9 =

Log2 7 - log2 11 =

Power Examples:

Log2 32 =

Ln (st)7 =

Log4 √51 =

4Log2 3 =

Power Examples:

Log3 (5) =

Log4 x =

Log2 7x =

Concept Summary:

Properties of Logarithmic Functions:

– logb (A ∙ B) = logb A + logb B

– logb (A / B) = logb A – logb B

– logb (A)p = p∙logb A

– Change bases:

– Remember: logb (x + y) ≠ logb x + logb y

Homework: pg 662-665; problems 1, 2, 6-9

Activity 5.14: Prison Growth

SOLs: None

Objectives: Students will be able to:

Solve exponential equations both graphically and algebraically

Vocabulary: None new

Key Concept:

Graphical Approach

1. Let Y1 = abx

2. Let Y2 = c

3. Find the intersection point of the two graphs

Algebraic Approach

1. Rewrite equation into form: bx = c/a (all positive)

2. Take ln or log of both sides: ln bx = ln (c/a)

3. Use properties of logs to simplify x ln b = ln c – ln a

4. Solve for the variable x = (ln c – ln a) / ln b

Activity:

Your sister is a criminal justice major at WCC. The following statistics appeared in one of her required readings relating to the inmate population of US federal prisons (population in thousands).

Year |1975 |1979 |1986 |1990 |1994 |1998 |2000 |2003 | |Population |20.1 |21.5 |31.8 |47.8 |76.2 |95.5 |112.3 |158.0 | |She asks you to help analyze the prison growth situation for a project in her criminology course.

Enter the data and plot using STATPLOT.

1. What form does the graph look like?

2. Use an exponential regression model on your calculator to estimate the total inmate population. Y = abx

3. Graph the STATPLOT and the Y1 = abx on the same graph

4. Use the model to estimate the prison population in 2010

5. Solve 11.36(1.080)t = 180 algebraically and then graphically.

Radioactive Decay:

Radioactive substances, such as uranium-235, stontium-90, iodine-131, and carbon-14, decay continuously with time. If P0 represents the original amount of a radioactive substance, then the amount P present after a time t (usually measured in years) is modeled by

P = P0ekt

where k represents the rate of continuous decay

Example 1: One type of uranium decays at a rate of 0.35% per day. If 40 pounds of this uranium was found today, how much will be left after 90 days?

Example 2: Strontium-90 decays at a rate of 2.4% per year. If 10 grams of it were initially present, how much will be left after 20 years?

Example 3: Strontium-90 decays at a rate of 2.4% per year. If 10 grams of it were initially present, how long until half of it is gone? (This is called half-life, a very important term with radioactive material)

Concept Summary:

– Solving exponential problems in form abx = c

1. Isolate exponential factor on one side of equation

2. Take the log (or ln) of both sides of equation

3. Apply log property: log bx = x log b to remove variable as an exponent

4. Solve the resulting equation for the variable

Homework: pg 669 – 72; problems 1, 4-10

Activity 5.15: Frequency and Pitch

SOLs: None

Objectives: Students will be able to:

Solve logarithmic equations both graphically and algebraically

Vocabulary:

None new

Key Concept:

Graphical Approach

1. Let Y1 = logb (f(x))

2. Let Y2 = c

3. Find the intersection point of the two graphs

Algebraic Approach

1. Rewrite equation into form: logb (f(x)) = c (all positive)

2. Rewrite step 1 in exponential form: f(x) = bc

3. Solve the resulting equation from step 2 algebraically

4. Check solution in the original equation

Activity: Raising a musical note one octave has the effect of doubling the pitch, or frequency, of the sound. However, you do not perceive the note to sound “twice as high,” as you might predict. Perceived pitch is given by the function

P(f) = 2410 log (0.0016f + 1)

where P is the perceived pitch in mels (units of pitch) and f is the frequency in hertz.

Graph the function

What is the perceived pitch, P, for an input of 10,000 hertz?

Write an equation that can be used to determine what frequency, f, gives an output of 2000 mels.

Solve it using the graphing approach

Solve the equation, 2410 log (0.0016f + 1) = 2000 using an algebraic approach

Use an algebraic approach to determine the frequency, f, that produces a perceived pitch of 3000 mels.

Concept Summary:

– Graphical Solution

1. Y1 = log function; Y2 = constant value; Graph and find intersection

– Algebraic Solution

1. Rewrite equation into form: logb (f(x)) = c (all positive)

2. Rewrite step 1 in exponential form: f(x) = bc

3. Solve the resulting equation from step 2 algebraically and Check solution in the original equation

Homework: pg 675 – 76; problems 1-6, 8

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