Chapter 9: Exponents, Exponents & More Exponents

Index:

A: Percent Review

B: Percent Increase and Decrease

C: Simplifying Expressions Involving Exponents D: Zero and Negative Exponents E: Exponential Growth and Decay F: Introduction to Exponential Functions G: Modeling with Exponential Functions

H: Linear Versus Exponential

Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Percent Review 9A - Classwork One of the major topics that you studied in 7th Grade Common Core Math was the concept of a percent. Quite possibly, percents are the most applied and misunderstood concepts in mathematics. In today’s lesson, we will review the major ideas and problems dealing with percents. The main idea of percents is as follows.

Examples: A) is of what number?

Exercise #1: Jonathan is getting a raise from $12.50 per hour to $14.75 per hour. His supervisor, Makayla, got a raise from $22.00 per hour to $25.30 per hour.

A percent always compares two quantities as a proportional relationship out of 100.

(a) How much in terms of dollars per hour is Jonathan’s salary going up?

(b) By what percent is Jonathan’s salary increasing?

(c) Who received the larger increase in salary in terms of dollars per hour?

(d) Who received the larger percent increase in salary?

You can pretty much always solve percent problems by setting up proportional equations that involve 100. Exercise #2: Gabe is buying a pair of jeans at a local store that are priced at $45. He knows that the county he lives in has an 8% sales tax added onto the list price. If Gabe has a gift card for $50, will it cover the cost of the jeans and the tax? Show work to justify your answer. Exercise #3: The population of deer in a forest preserve is predicted to decline by 5% this year. If the current population is 560, what population is predicted for next year? What percent of the deer will remain? Although it is convenient to solve percent problems using ratios, it is critical that you learn a different method. Some of you may have seen this before, if you’ve had teachers show it to you. If not, please make sure to understand what follows. Exercise #4: The United States population is roughly 314 million people (314,000,000). The workforce participation rate, defined as the percent of the population working or looking for work, is 62.8%. The unemployment rate is the percent of the workforce that is looking for work, but cannot find it.

(a) How many people are working or looking for work in the United States?

(b) If the unemployment rate is currently 6.8%, then how many people are unemployed to the nearest hundred thousand.

Know this important method for finding percents of totals: This “quick” way of finding percents of totals is a skill that you must become fluent with. Let’s get some practice with it in the next exercise. Exercise #5: Find each of the following. Write down the product that you use to find your answer.

(a) 20% of 85 (b) 12% of 200 (c) 6% of 550

(d) 4.5% of 120 (e) 36% of 96 (f) 14

2 % of 350

Exercise #6: Which of the following calculations would find 8.5% of 250?

(1) 85 250 (3) 0.85 250

(2) 8.5 250 (4) 0.085 250

FINDING PERCENTS BY MULTIPLYING

To find p% of a total, T, simply find the product: 100

pT . Often

100

p is expressed as a decimal.

Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Percent Review 9A - Homework 1. Evaluate the following percent problems by setting up and solving a ratio like we did in Exercises #1 through #3 in the lesson. (a) Find 7% of 280 (b) Find 12% of 300 (c) Find 2% of $1250 2. Find each of the following by a single multiplication problem (like what we did in Exercise #5 from the lesson). Write down the product that you use in your calculation. (a) Find 6% of 350 (b) Find 25% 80 (c) Find 15% of $35.00 3. Find each of the following by a single multiplication problem (like what we did in Exercise #5 from the lesson). These are trickier than #2. If needed, take the percent and divide it by 100 on your calculator to determine what to multiply by. Write down the product that you use in your calculation. Do not round your final answers. (a) 3.2% of 360 (b) 2.7% of 90 (c) 12.8% of 240 (d) 0.8% of 450 (e) 0.5% of 500 (f) 0.25% of 320 4. If x represents 2.8% of 270, then which of the following equation would not result in the correct value for x?

(1) 2.8

270 100

x (3) 0.028 270x

(2) 2.8

270100

x (4) 0.28 270x

5. Prestel currently makes $8.50 per hour. His boss has promised him a 15% raise in his hourly earnings.

6. Imani’s rent increased from $560 per month to $600 per month. Her friend, Ariana, had her rent increase from $825 to $875. Who had the larger percent increase in their rent? Remember to set up your ratios using the original rent. See Exercise #1 from the lesson if you want to see a similar problem. 7. Consider the following problem. Let’s say that I want to leave a 15% tip on a meal that cost $35.

8. Niko had his savings increase by 5% this year. He started with $350 in his account and calculated how much he had at the end of the year by using the following sets of calculations: Find a single number that Niko could have multiplied his starting amount of $350 by to get $367.50 by solving the equation below for r. Why does this number make sense?

(a) Calculate 15% of $8.50. Why can’t Prestel get exactly a 15% raise? What would his boss actually give him?

(b) After the raise, what is Prestel’s new salary? Show the calculation that leads to your answer.

$350 0.05 $17.50 $350 $17.50 $367.50

350 367.50r

(a) Find the tip by setting up a ratio involving 100. (b) Find the tip by doing a single multiplication problem. Why is this the same as (a)?

Review Section: 9.

Name:_____________________________________________________ Date:_________________ Period:_________ Algebra I Percent Review 9A HW 1. (a) 19.6 (b) 36 (c) $25 2. (21 (b) 20 (c) $5.25 3. (a) 11.52 (b) 2.43 (c) 30.72 (d) 3.6 (e) 2.5 (f) .8 4. (4) 5. (a) $1.28 with an appropriate explanation (b) $9.78 6. Imani; with appropriate work 7. (a) $5.25 (b) $5.25, with appropriate explanation 8. r=1.05; with a proper explanation 9. (2)

Homework Answers

Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Percent Increase/Decrease 9B - Classwork In the last lesson we learned a quick way to find a percent of a total. In this lesson, we would like to develop and understand methods for increasing and decreasing a number by a certain percent. We will start with the increasing case. Exercise #1: Agronomists are studying how quickly the population of an invasive species of beetle will increase in a controlled farm setting. They calculate that the population is increasing at a steady rate of 6% per week. At the beginning of the week, the population was 350 beetles. (a)Find the population of beetles a week later by first finding 6% of 350 and adding it to the original population.

The ability to increase a total by a certain percent using this method is important. Get some fluency with it in the next problem. Exercise #2: Find the result of each of the following. Many of your answers will involve decimals. Do not round. (a) Increasing 440 by 12% (b) Increasing 68 by 8% (c) Increasing 120 by 3.5% Exercise #3: Adriana has a saving account that promises to increase her balance by 2.5% per year. If she deposits $720 in it at the beginning of the year, which of the following would be her balance at the end of the year if she does not withdraw or deposit any additional money?

(1) $900 (3) $738 (2) $842 (4) $756

(b) Find the population of beetles a week later by a single multiplication. Why does this work?

(c) What will be the beetle population after two weeks?

We would also like to work with decreasing by a certain percent. This will follow a similar, if not identical, pattern to the increasing case. Again, let’s understand what is going on with an introductory problem. Exercise #4: A cup of coffee is cooling down such that its temperature is decreasing at a constant rate of 8% per

minute. Let’s say the coffee starts at a temperature of 200 F . (a) Find its temperature after one minute by finding 8% of 200 and then subtracting.

Decreasing by a certain percent is an important skill to be fluent with as well. It often is harder for students because they need to think about what percent remains. Exercise #5: The enrollment of students at a school is decreasing at a constant rate of 5% per year.

Exercise #6: Find the result of each of the following. Many of your answers will involve decimals. Do not round. Write the calculations you use to find yours answers. (a) Decrease 620 by 10% (b) Decrease $22.50 by 8% (c) Decrease 122 by 3.5% Exercise #7: The cost of gasoline has decreased recently by 4.5%. If it started at $3.80 per gallon, which of the following is its price after the decrease?

(1) $3.63/gal (3) $3.22/gal (2) $3.72/gal (4) $3.97/gal

(b) Find its temperature after one minute by finding a single product. How can you interpret this in terms of the 8%?

(c) What will the temperature of the coffee be after two minutes? Round to the nearest degree.

(a) What percent remains after one year? (b) If the population this year is 2300, what will its population be next year? Do in a single calculation.

Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Percent Increase/Decrease 9B - Homework 1. Perform each of the following calculations using a single multiplication. Show the product that you use to find your final answer. Do not round your final answers. (a) Increase 350 by 5% (b) Increase 120 by 10% (c) Increase 34 by 2%

(d) Increase $450 by 3.5% (e) Increase $1,300 by 12

6 % (f) Increase 2,698 by 34

2 %

2.Perform each of the following calculations using a single multiplication. Show the product that you use to find your final answer. Do not round your final answers. (a) Decrease 160 by 10% (b) Decrease 450 by 6% (c) Decrease 122,000 by 12% (d) Decrease $1,820 by 3% (e) Decrease $12,500 by 15% (f) Decrease $4.50 by 8% 3. A population of bacteria is growing at a rate of 20% per hour. If the population starts at 320, what is it an hour later? (1) 360 (3) 372 (2) 356 (4) 384

4. The price of oil, in dollars per barrel, declined last week by 3.5%. If it started the week at $102.00 per barrel, at what per barrel price did it end the week? (1) $98.43 (3) $99.12 (2) $98.50 (4) $100.56 5. A savings account grows by 3% per year. Sofia places $500 in the account at the beginning of the year.

(c)Did the amount of money in Sofia’s account grow faster the first year or the second year? Explain how you arrived at your answer. Use proper units in your explanation.

6. An environmental firm is removing pollution from a river bed. They find that the pollution is decreasing at a rate of 5% per month, as measured by parts per million. The pollution starts off at a level of 360 parts per million (ppm).

(a) How much does Sofia have at the end of the year once her $500 has been increased by 3%?

(b) How much does Sofia have at the end of the second year based on increasing your answer from part (a) by 3%?

(a) What is the pollution level one month after they begin clean-up? Use appropriate units.

(b) What is the level two months after? Use appropriate units.

Percents build on one another in strange ways. It would seem that if you increased a number by 5% and then increased its result by 5% more, the overall increase would be 10%. Let’s see if it really is that simple. 7. Let’s do exactly this with the easiest number to handle in percents. (a) Increase 100 by 5% (b) Increase your result form (a) by 5%. (c) What was the overall percent increase of the number 100? Why is it not 10%? Review Section: 8. Based on the graph of the function shown below, answer the following questions. (a) Evaluate each of the following. Illustrate with a point on the graph. (b) What values of x that solve the equation ? (c) Name the coordinate of the absolute minimum. (d) Name an interval that is increasing. 9. Determine the smallest integer that makes true.

Name:_____________________________________________________ Date:_________________ Period:_________ Algebra I Percent Increase/Decrease 9B HW 1. (a) 367.5 (b) 132 (c) 34.68 (d) 465.75 (e) $1,384.50 (f) 2,772.195 2. (a) 144 (b) 423 (c) 107,360 (d) $1,765.40 (e) 10,625 (f) $4.14 3. (4) 4. (1) 5. (a) $515 (b) $530.45 (c) 2nd year (show proper work) 6. (a) 342 ppm (b) 324.9 ppm 7. (a) 105 (b) 110.25 (c) 10.25% (show work and explain) 8. (a) -3 and 4 (b) {-3,1,5,7} (c) (-1,-4) (d) 9. 0

Homework Answers

Name _________________________________________________ Date _____________________ Algebra 1 – PD ___ Simplifying Expressions 9C - Classwork There are many situations in science, engineering and other fields where a process is governed by repeatedly multiplying (or dividing) by the same quantity. Repeated multiplication (and division) is represented by exponents. Exercise #1: Each of the following problems involves basic exponent ideas. Answer each to review your previous knowledge.

(a) Represent 36 as an extended product. Do not (b) If is written in the form of , what is evaluate the product. the value of n? Write extended products if you

don’t remember the Exponent Rule.

(c) If the expression is written in the (d) Rewrite the product as an form , what is the value of equivalent expression in simplest exponential form? Identify the Exponential Rules:

Exponent Rule #1:

Exponent Rule #2:

Exponent Rule #3:

When multiplying Monomials: Coefficients: __________________ the coefficients. Variables: When multiplying the variables of monomials you ____________________ the base and

____________ the exponents (Remember if there is no exponent written, the exponent is 1.)

Exercise #2: Using the above exponential rules, simply the following expressions. (a) (b) (c) (d) (e) (f) If the length of a rectangle is m and its width is m, what is its area

written in scientific notation?

We also would like to be able to write simpler equivalent expression involving ratios (or division problems) involving exponents. This all comes down to your ability to “unmultiply” fractions. Identify the Exponential Rules:

Exponent Rule #1:

Exponent Rule #2:

Exponent Rule #3:

Exponent Rule #4:

When dividing Monomials: Coefficients: __________________ the coefficients. Variables: When dividing the variables of monomials you ____________________ the base and

____________ the exponents (Remember if there is no exponent written, the exponent is 1.)

Exercise #3: Using the above exponential rules, simply the following expressions.

(a)

(b)

(c)

(d)

Name _________________________________________________ Date _____________________ Algebra 1 – PD ___ Simplifying Expressions 9C – Homework _____ 1.) Which of the following is equivalent to [1] [3] [2] [4] _____ 2.) If the expression was written in form, which of the following would be the sum of and ? [1] 20 [3] 9 [2] 14 [4] 18 _____ 3.) A square field has a side length of meters. Which of the following is its area? [1] [3] [2] [4] 4.) Circle the reasons for each of the following manipulations used to simplify the product

Commutative Commutative Commutative or Associative or Associative or Exponent 5.) Using your four exponent rules, simplify each of the following expressions completely.

a.)

b.)

c.)

d.)

e.)

f.)

g.)

h.)

i.)

6.) Kris has incorrectly simplified the expression

as .

a.) Show using the value of that

b.) What is the correct simplification?

and are not equivalent. Review Section: _____ 7.) 8.) Graph the function on the set of axes. Explain how the graph of has changed from the related graph of .

Name _________________________________________________ Date _____________________ Period ____ Algebra Simplifying Expressions 9C HW 1.) 2 2.) 1 3.) 3 4.) Commutative, Associative, Exponential

5.) a)

b)

c)

d)

e)

f)

g)

h)

i)

6.) a)

Therefore they are not equivalent.

b) 7.) 2 8.) Graph is the graph of shifted to the right three units.

Homework Answers

Name _________________________________________________ Date _____________________ Algebra 1 – Pd___ Zero & Negative Exponents 9D - Classwork In math, people often invent ways to extend concepts to areas that might not make sense at first. Pretty much everyone can understand what means, because they understand that it represents multiplying by the number 2 a total of 3 times. Two special cases are when terms are raised to the zero power or raised to a negative power. Zero Power: Anything raised to the power of zero is equal to one. Ex: ; Negative Powers: Take the reciprocal of the base (the base is the term that is raised to the power), then make the exponent positive and simplify as necessary.

Ex:

We want the pattern of positive, integer powers to extend to zero exponents and negative, integer exponents. We can now define zero and negative exponents as follows.

Zero & Negative Exponents 1.) Zero Exponents:

as long as

2.) Negative Exponents:

Exercise #2: Which of the following is not equivalent to ?

[1]

[3]

[2]

[4]

Exercise #3: If , then which of the following is the value of Show the work that leads to your answer. Remember exponents always come before multiplication.

[1]

[3]

[2]

[4]

Exercise #4: Using all your known exponent rules, simplify each of the following completely.

(a)

(b)

(c)

(d)

So, we now see that the subtraction rule for exponents is consistent with negative and zero exponents. For now, we just want to be comfortable that negative exponents indicate division and positive exponents indicate multiplication. Exercise #5: Consider the exponential function . Find each of the following without the use of your calculator. (a) (b) (c)

Name _________________________________________________ Date __________________________ Algebra I CC – Pd ___ Zero & Negative Exponents 9D Homework 1.) Rewrite each of the following as equivalent expressions without the use of negative or zero exponents. Remember your order of operations. (a) (b) (c) (d) (e) (f) _____ 2.) Which of the following is not equivalent to

[1]

[3] 0.125

[2] [4]

_____ 3.) If , then which of the following represents the value of [1] [3] [2] [4] _____ 4.) If the expression is evaluated when , the result would be: [1] 1 [3] 7 [2] 0 [4] 4

_____ 5.) The numerical expression

is equivalent to:

[1]

[3]

[2] [4]

6.) Write each of the following in the form , where is either a positive or negative integer.

(a)

(b)

(c)

7.) The number of people, n, who know a rumor can be modeled using the equation , where is the number of days since Monday. (a) Explain why . What does this (b) What is the value of ? What does this represent in terms of the situation modeled? represent in terms of the situation modeled? _____ 8.) Consider the function . When the value of x is increased by , the output is: [1] multiplied by 3 [3] multiplied by [2] divided by 3 [4] divided by Review Section: _____ 9.)

_____ 10.) _____ 11.) Consider the function . When the value of x is increased by , the output is: [1] multiplied by 7 [3] multiplied by [2] divided by [4] divided by 7

Name _________________________________________________ Date _____________________ Period ____ Algebra Zero & Negative Exponents 9D HW

1.) a)

b) 1 c)

d) 4 e)1 f)

2.) 2 3.) 3 4.) 2 5.) 1

6.) a) b) c)

7.) a) represents the fact that 20 people knew the rumor on Monday. b) represents the fact that 5 people knew the rumor two days before Monday (Saturday). 8.) 2 9.) 1 10.) 3 11.) 4

Homework Answers

Name _________________________________________________ Date _____________________ Algebra 1 – Pd___ Exponential Growth/ Decay 9E - Classwork There are many things in the real world that grow faster as they grow larger or decrease slower as they get smaller. These types of situations, loosely speaking, are known as exponential growth (and decay in the case of decreasing). In today’s lesson, we will look at both growth and decay. Exercise #1: The number of people who have heard a rumor often grows exponentially. Consider a rumor that starts with 3 people and where the number of people who have heard it doubles each day that it spreads. (a) Why does it make sense that the number of (b) Fill in the table below for the number of people who have heard a rumor would grow people, N, who knew the rumor after it has exponentially? spread a certain number of days, d. Exponential Functions: Standard form: where a is the initial amount (y-intercept) b is the rate of change x is time More specifically for growth we use: where r = % rate of change for decay we use: and should be converted to a decimal. (c) How many people would know the rumor after 20 days? (d) Exponential growth can be very fast. How many days will it take for the number of people knowing the rumor to surpass the population of the United States, which is approximately 315 million people? Show calculations that support your answer.

Example:

when a > 0 and the b is between 0 and 1, the graph will

be decreasing (decaying).

For this example, each time x is increased by 1, y

decreases to one half of its previous value.

Such a situation is called Exponential Decay.

Example:

when a > 0 and the b is greater than 1, the graph

will be increasing (growing).

For this example, each time x is increased by 1, y

increases by a factor of 2.

Such a situation is called Exponential Growth.

Exercise#2: A bank account balance, b, for an account starting with s dollars, earning an annual interest rate, r,

and left untouched for n years can be calculated as (an exponential growth formula). Find a

bank account balance to the nearest dollar, if the account starts with $100, has an annual rate of 4%, and the

money left in the account for 12 years.

Exercise#3: In 1995, there were 685 rabbits in Central Park. The population decreased by 12% each year.

How many rabbits were in Central Park in 2002? In 1985, there were 285 cell phone subscribers in the small

town of Centerville.

Exercise#4: The number of subscribers increased by 75% per year after 1985. How many cell phone

subscribers were in Centerville in 1994?

Exercise#5: Daniel’s Print Shop purchased a new printer for $35,000. Each year it depreciates (loses value) at

a rate of 5%. What will its approximate value be at the end of the fourth year?

Exercise#6: The amount A, in milligrams, of a 10-milligram dose of a drug remaining in the body after t hours

is given by the formula . What is the percent rate of change?

Name _________________________________________________ Date _____________________ Algebra 1 – Pd___ Exponential Growth /Decay 9E Homework 1.) Sheba opened a retirement account with $36,500. Her account grew at a rate of 7% per year compounded

annually. She made no deposits or withdrawals on the account. At the end of 20 years, what was the account

worth, to the nearest dollar?

2) Cassandra bought an antique dresser for $500. If the value of her dresser increases 6% annually, what will be

the value of Cassandra's dresser at the end of 3 years to the nearest dollar? 3) You decide to buy a boat that costs $8550. The normal depreciation for such a boat is 17% per year. If you

pay for the boat with a 5 year loan, how much less will the boat be worth after you have paid off the loan?

4) The current student population of the Brentwood Student Center is 2,000. The enrollment at the center

increases at a rate of 4% each year. To the nearest whole number, what will the student population be closest to

in 3 years'?

_____5) The projected worth (in millions of dollars) of a large company is modeled by the equation . The variable x represents the number of years since 1997. What is the projected annual percent of

growth, and what should the company be worth in 2005?

[1] 21%; $273.06 million [3] 11%; $566.92 million

[2] 21%; $629.28 million [4] 11%; $510.74 million Review Section: _____ 6.) _____ 7.) 8.)

Name _________________________________________________ Date _____________________ Period ____ Algebra Exponential Growth 9E HW 1.) $141,243 2.) $596 3.) $5182.12 4) 2,249 students 5) (3) 6) (1) 7.) (2) 8) because the x value will repeat which will create a relation.

Homework Answers

Name _________________________________________________ Date _____________________ Algebra 1 – Pd___ Exponential Functions 9F – Classwork Before we begin with the next lesson, calculate the slope of the line that would be formed by (3,4)(-9,-4).

So far we have concentrated on linear functions which are characterized by having a constant rate of change. In

the last lesson, we looked at exponential growth and decay. In this lesson we will more formally introduce the

concept of an exponential function.

Exercise #1: Consider the exponential function . Answer the following. (a) Evaluate each of the following and indicate what point must lie on the graph of based on each: (i) (ii) (iii) (b) Calculate the average rate of change of f over (c) Calculate the average rate of change over the the interval . interval (d) What does comparing answers from (b) and (c) (e) Using your calculator, draw a sketch of this tell you about this function? Explain. function on the axes below using the

window indicated.

Exponential functions are all about multiplication. The basic form of an exponential function is given below.

Exercise #2: Consider the function

(a) Evaluate . What point does this indicate (b) Without the use of your calculator, on the graph of g? determine the values of and (c) Using your graphing calculator, sketch a graph (d) Why is this exponential function always of this function using the WINDOW decreasing while the one in Exercise #1 and . Mark the y-intercept. is always increasing? Exercise #3: For each of the following exponential functions, give its y-intercept and tell whether it is increase or decreasing.

(a)

(b) (c)

The equations of exponential functions are relatively easy to determine, if you understand this lesson so far. See what you can do in the next exercise. Exercise #4: Find the equation of the exponential function, in form, for the function given in the

table below. Show or explain your thinking.

Name _________________________________________________ Date _____________________ Algebra 1 – Pd___ Exponential Functions 9F Homework

Increasing VS Decreasing Exponentials

will increase if __________________________

will decrease if __________________________ 1.) Consider the exponential function (a) Find the value of . What point does this (b) Is this an increasing or decreasing represent on the graph of ? exponential function? How can you tell based

on its equation?

(c) Is this function’s average rate of change over the (d) Using your calculator, sketch a graph of this interval greater or less than that of the function on the axes shown below. Use the linear function ? Justify. window indicated. Mark the y-intercept. _____ 2.) Which of the following is a decreasing exponential function whose y-intercept is 20?

[1]

[3]

[2]

[4]

_____ 3.) Which of the following functions would best describe the data in the table? [1] [3] [2] [4] 4.) Graphing a basic exponential can be challenging because of how quickly they grow (or decay). In this exercise, we will graph one of the most basic. (a) Evaluate each of the following and state the coordinate point that occurs on the graph of based on the calculation. (b) Evaluate each of the following. Remember your facts about negative exponents and give the point on the graph of . (c) Using the points you found in (a) and (b), graph this function for the domain interval . 5.) Classify each of the following exponential functions as either increasing or decreasing and give the value of their y-intercepts.

(a) (b)

(c)

_____ 6.) Which of the following could be the equation of the exponential function shown graphed below? [1] [3] [2] [4] Review Section: _____ 7.) 8.)

Name _________________________________________________ Date _____________________ Period ____ Algebra Exponential Functions 9F HW 1.) a) This point is the y-intercept. b) This is an increasing exponential function because its base value, b = 2, is greater than 1.

c) Average Rate of Change:

The function has a greater average rate of change over this interval. d) graph 2.) 2 3.) 4 4.) a)

b)

c) Graph 5.) a) Increasing function y-intercept = 125 b) Decreasing function y-intercept = 22 c) Increasing function y-intercept = 256 6.) 2 7.) 4 8.) Graph

Homework Answers

Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Exponential Models Based on Percent Growth 9G - Classwork There are many examples of growth in the real world that occur at a constant percent rate. These phenomena give rise to exponential functions. These functions will be easy to build and understand if you felt comfortable with the last lesson on percent growth and decay. Exercise #1: A population of fruit flies is growing at a constant rate of 6% per hour. The population starts, at 0t with 28 flies.

Exponential growth is fairly easy to model and fairly easy to interpret in the model. Consider the following example.

Exercise #2: If the savings in a bank account can be modeled by the function 250 1.045t

S t . Which of the

following is true?

(1) The initial amount deposited was $250 and the interest earned is 45%. (2) The initial amount deposited was $2.50 and the interest rate is 4.5%. (3) The initial amount deposited was $250 and the interest rate is 4.5%. (4) The initial amount deposited was $2.50 and the interest rate is 45%.

(a) Using what we learned in the last lesson, determine the population after each of the following amounts of time. Show the calculation you use as repeated multiplication.

1 hr

2 hr

3 hr

t P

t P

t P

(b) Based on (a), find a formula that models the population, P, as a function of the time in hours, t.

(c) What is the value of 24P ?

(d) What does the calculation you made in (c) represent about the fly population? State the range of the population function over the domain interval 0 24t .

(e) Using your calculator, sketch a graph of this function over the interval 0 24t and

0 120P . Mark the y-intercept.

P

t (hrs)

We should also be able to model exponentially decreasing phenomena based on what we learned in the last lesson about percent decrease. Remember to always model based on the percent that remains. Exercise #3: As water drains out of a pool, the depth of the water decreases at a constant percent rate of 20% per hour. The depth of the water, when the draining begins, is 12 feet.

(a) As in #1, find the depth, D, of the water in the pool after each of the following times, t.

1 hr

2 hr

3 hr

t D

t D

t D

(b) Based on (a), create an equation that gives the depth, D, of the water in the pool as a function of the time in hours it has been draining, t.

(c) Using your calculator, sketch a graph of your function over the interval 0 20t and

0 15D . Mark the y-intercept with its value.

(d) It’s safe to cover the pool after it reaches a depth of 1 foot or less. What is the minimum number of whole hours that we should wait to cover the pool? Explain how you found your answer.

D

t (hrs)

D

t (hrs)

Exercise #4: An oil spill is spreading such that its area is given by the exponential function 250 1.15t

A t ,

where A is the area in square feet and t is the time that has elapsed in days.

**WHEN YOU ARE FINISHED WITH THIS QUESTION, CLICK ZOOM 6 TO PUT TABLE BACK TO “NORMAL”** Exercise #5: If $350 is placed in a savings account that earns 3.5% interest applied once a year, then how much would the savings account be worth after 10 years?

(1) $522.88 (3) $472.50 (2) $426.34 (4) $493.71

(a) How large was the oil spill initially, i.e. at 0t ? (b) By what percent is the oil spill increasing each hour?

(c) Sketch a graph of the area of the oil spill over the interval 0 30t and 0 5000A using your calculator. Label the y-intercept.

(d) After how many days will the oil spill reach a size of 3,000 square feet? Round to the nearest tenth. Solve graphically using the INTERSECT

COMMAND on your calculator.

(e) What is the average rate of change of A t over

the interval 0 10t ? Include proper units and don’t round.

Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Exponential Models Based on Percent Growth 9G - Homework

Growth Decay initial amount before measuring growth/decay growth/decay rate (often a percent and must be converted to a decimal) number of time intervals that have passed

1. In 1995, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 45% per year after 1995. How many cell phone subscribers were in Centerville in 2006? 2. Daniel’s Print Shop purchased a new printer for $35,000. Each year it depreciates at a rate of 5%. What will its approximate value be at the end of the fourth year?

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3. The breakdown of a sample of a chemical compound is represented by the function where represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent.

4. If a flock of ducks is growing by 6% per year and starts with a population of 68, how many ducks will be in the flock after 10 years? (1) 109 (3) 122 (2) 198 (4) 408 5. A bank account earns interest at a rate of 3.5% per year (in other words it increases in value by that percent) and starts with a balance of $350. Which of the following equations would give the account’s worth, W, as a function of the number of years, y, it has been gaining interest?

(1) 350 1.035y

W (3) 1.035 350W y

(2) 350 0.35y

W (4) 1.35 350W y

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6. Newton’s Law of Cooling can be used to predict the temperature of a cooling liquid in a room that is at a certain steady temperature. We are going to model the temperature of a cooling cup of coffee. The Fahrenheit temperature

of a cup of coffee, T, in a room that is at a 72 F is given as a function of the number of minutes, m, it has been cooling by:

(a) Find 0T and using proper units, give a

physical interpretation of your answer.

(b) What does the coefficient of 114 represent in terms of the situation being modeled?

(c) By what percent does the difference between the temperature of the coffee and the temperature of the room decrease each minute?

(d) I like my coffee when it is a nice temperature of

around 100 F . How long should I wait?

114 0.86 72m

T m

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Review Section: 7. On the set of axes below, graph the inequality . 8. 9.

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Name:_____________________________________________________ Date:_________________ Period:_________ Algebra I Exponential Models Based on Percent Growth 9G HW 1. 16,978 cell phone subscribers 2. $28,507.72 3. give a proper interpretation 4. (3) 5. (1) 6. (a) (b) proper explanation (c) 14% (d) 9-10 minutes 7. Graph with complete work and proper shade 8. (3) 9. (3)

Homework Answers

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Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Linear vs. Exponential 9H - Classwork Linear and exponential functions share many characteristics. This is because they are based on two different, but similar, sets of principles. Exercise #1: The two tables below represent a linear function and an exponential function. Which is which? Explain how you arrive at your answer. Exercise #2: Find equations in standard form for each of the functions from Exercise #1.

(a) Table 1 (b) Table 2 It is interesting that linear and exponential functions are ones where two points on the curve will always determine the equation of the curve.

Exercise #3: Consider the two points 0,12 and 1, 3 . Create a linear equation that passes through these points

in y mx b form and an exponential equation in x

y a b form that also passes through them. Then, using

your calculator, graph both using a window of 2 2x and 5 15y .

LINEAR VERSUS EXPONENTIAL

Linear functions are based on repeatedly adding the same amount (the slope).

Exponential functions are based on repeatedly multiplying by the same amount (the base).

TABLE 1

x 0 1 2 3 4

y 5 10 20 40 80

TABLE 2

x 0 1 2 3 4

y 8 11 14 17 20

y

x

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Recall that linear functions have a constant average rate of change (slope). That’s, of course, why they have a constant amount added for every constant change in x. Let’s examine the average rate of change for an increasing exponential.

Exercise #4: The exponential function 4 2x

f x is shown partially in the table below. Find the average rate

of change over the various intervals given. This should be relatively simple because 1x .

(a) 0 1x (b) 1 2x (c) 2 3x (d) 3 4x (e) What is clearly happening to the average rate of change as x gets larger? The fact that the slope of an increasing exponential is always increasing has an interesting consequence.

Exercise #5: Consider the linear function 20 5y x and the exponential function 5 2x

y . Both of these

functions have a y-intercept of 5, so “start” in the same location.

(c) Although the line appears to rise more quickly than the exponential, at first, the exponential eventually catches up and surpasses the linear. Why will an increasing exponential function always catch up with an increasing linear function?

x 0 1 2 3 4

y 4 8 16 32 64

(a) Using your calculator, sketch these two curves on the axes below for the indicated window. Label each with its equation.

(b) Again, using your calculator, sketch these two curves on the axes below for the indicated window. Label each with its equation.

y

x

30

5

3

y

x

150

5

6

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Name:_____________________________________________________ Date:___________________________ Algebra I CC – Pd______ Linear vs. Exponential 9H - Homework 1. For each of the following problems a table of values is given where 1x . For each, first determine if the table

represents a linear function, of the form y mx b , or an exponential function, of the form x

y a b . Then,

write its equation. (a) (b) (c) (d) (e) (f) 2. The data shown in the table below represents either a linear or an exponential function. Which of the equations below best models this data set?

(1) 5 2x

y (3) 2 10y x

(2) 10 2x

y (4) 10 5y x

x 1 0 1 2 3

y 4 7 10 13 16

Type: ____________________________

Equation: _________________________

x 0 1 2 3 4

y 2 6 18 54 162

Type: ____________________________

Equation: _________________________

x 2 1 0 1 2

y 32 16 8 4 2

Type: ____________________________

Equation: _________________________

x 2 1 0 1 2

y 32 16 0 16 32

Type: ____________________________

Equation: _________________________

x 0 1 2 3 4

y 16 20 25 14

31 116

39

Type: ____________________________

Equation: _________________________

x 0 1 2 3 4

y 180 160 140 120 100

Type: ____________________________

Equation: _________________________

x 1 2 3 4

y 10 20 40 80

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3. Wildlife biologists are tracking the population of albino deer in an upstate New York forest preserve. They record the population every year since 2005, which they consider to be 0t . Their data is shown in the table below. (a) Although neither a linear nor an exponential function would model this data perfectly, justify why an exponential function would be a much better fit. Specifically, explain both why a linear function would not be a good fit while an exponential would be reasonable.

4. You can determine the equation of a line or the equation of an exponential given any two points that lie on these

curves. In this exercise we will pick two special points. Consider the points 0, 5 and 1,15 .

Year 2005 2006 2007 2008 2009 2010

t 0 1 2 3 4 5

Population 86 98 111 128 147 168

(b) Determine an equation for an exponential that

models this data set in the form t

P a b .

Justify both your values of a and b. Round b

to the nearest hundredth.

(c) Use your model to predict the population of deer in the year 2014.

(a) Write the equation of the line that passes

between these two points in y mx b form.

(b) Write the equation of the exponential that

passes between these two points in x

y a b

form.

(c) Using your calculator, sketch the two curves on the axes below. Label with their equations.

3 3

30

10

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5) Review Section: 6.

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Name:_____________________________________________________ Date:_________________ Period:_________ Algebra I Linear vs. Exponential 9H HW

1. (a) Linear; (b) Exponential; (c) Exponential;

(d) Linear; (e) Exponential; (f) Linear; 2. (1) 3. (a) Show evidence as to why neither work (b) (c) P=280 albino deer 4. (a) (b) (c) sketch the two curves 5. Exponential with appropriate explanation. 6. (a) (b) No (c) 10 cats

Homework Answers

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