youngstown city schools€¦ · web viewworksheet #2(attached on page 7) to reinforce properties...

YOUNGSTOWN CITY SCHOOLS

MATH: PRECALCULUS

UNIT 3: COMPOSITE FUNCTIONS AND INVERSES ( 4.5 WEEKS) 2013-2014

Synopsis: This unit focuses on combining functions; writing a composition of functions both as a skill and in real-life applications; finding inverses, and proving they are inverses by using composite functions. Students will also expand on the concept of inverses by examining logarithmic and exponential functions as inverses of each other and solving real-life problems involving these functions.

STANDARDSF.BF.1b Write a function that describes a relationship between two quantities; Combine standard function types using arithmetic operations. For example: build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.F.BF.1c Write a function that describes a relationship between two quantities; (+) Compose functions. For example: if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.F.BF.4b Find inverse functions. (+) verify by composition that one function is the inverse of anotherF.BF.4c Find inverse functions. (+) read values of an inverse function from a graph or a table, given that the function has an inverse.F.BF.4d Find inverse functions. (+) produce an invertible function from a non-invertible function by restricting the domain.F.BF.5 (+) understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents

MATH PRACTICES1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning

LITERACY STANDARDSL.1 Learn to read mathematical text (including textbooks, articles, problems, problem explanations )L.2 Communicate using correct mathematical terminology

MOTIVATION TEACHER NOTES1. Students will perform an experiment with either hot coffee or crushed iced. Have

students estimate how long it will take for the coffee to reach room temperature or how long for the ice to melt. Take either a cup of hot coffee or crushed ice, and record the temperature on a table every three minutes. Once all the data have been collected, place values on a graph. Perform a regression on the TI-Nspire calculator with the data, and then describe how this is an exponential function. Relate this to adding functions: exponential function added to a constant function. Simultaneously work on worksheet containing solving equations with logs (worksheet #4, attached on page 9; see T/L #1). (F.BF.1b, F.BF.5, MP.1, MP.2, MP.4, MP.5, MP.6, L.2)

2. Preview expectations for the end of the Unit

3. Have students set both personal and academic goals for this Unit

06/08/2013 YCS MATH PRE-CALC UNIT 3 COMPOSITE FUNCTIONS AND INVERSES 2013-14 1

TEACHING-LEARNING TEACHER NOTESVocabulary:

inverse growth decay exponential composition invertible range domain logarithm compound interest compound continuously non-invertiblenatural logs extrapolate

Good web site for this unit: http://www.purplemath.com/modules/fcncomp.htm

1. Review laws of exponents using worksheet #1(attached on page 6.)

2. Review definition of a log (page 719 in the textbook) and the properties of logs (page 720 in the textbook), including the change of base formula on page 728, then work problems on p. 723 and 724 and problems 34 through 39 on page 731 in the textbook.

3. Next progress to the definition of natural logs. Ask students to formulate the laws for natural logs and list on the board. Have them work on the worksheet #2(attached on page 7) to reinforce properties of natural logs. Work on finding the inverse of logs to show exponential function is its inverse. Teacher can have students work on this simultaneously while engaging in the motivational activity. Progress to worksheet #3 (attached on age 8) and worksheet #4 (attached on page 9 ) using the inverse functions to solve equations.

4. Work on adding functions h(t) = -4.9t2 +10t and s(t) = 30 meters. Add two functions together to find height of object at time t with a starting height of s(t). a) Have students find h(t) for particular times, t, b) find h(t) + s(t), c) vary s(t), the beginning height and find h(t) + s(t), d) discuss changes in the functions and graphs.

5. Example 1: At midnight, the police were called to the scene of a murder, where the coroner was to calculate the time of death. The formula is T(t) = A(0) + (T(0) – A(0))*e-rt, where T(t) is the temperature of the body when found, A(0) is the initial air temperature, T(0) is the original temperature of the body when found and t is the time. The air temperature was constantly 680F during the entire time, the temperature of the body was 850F and r = 0.5.

85 = 68 + (98.6 – 68) * e-0.5t

When solving this, the time is 1.18 hours, so he died approximately 1 hour and 10 minutes ago or 10:50 P.M. Look at the graphical representation of T(t), discuss the asymptote, y = 68 and the relationship between it and the function T(t)

t

T(t)

The following web site has an additional example of a cooling body temperature problem: www2.southeastern.edu/Grants/LASIP/lessons/dhlp.htmIf students understand this easily, teacher could discuss the method of finding r.(F.BF.1b, F.BF.5, MP.1, MP.2, MP.4, MP.5, MP.6, L.2)

6. Composition of functions: start with the example in Standard F.BF.1c. Example #1: If T(h) (changed from T(y)) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Write the equation for T(h(t)) where T(h) = 2h + 6, and h(t) = -0.003t2 - 0.6t + 30 (answer T(h(t)) = 2(-0.003t2 -


68

98.6

http://www.purplemath.com/modules/fcncomp.htm

TEACHING-LEARNING TEACHER NOTES0.6t + 30)+6. Then ask students to find the temperature of the balloon at a specific time, t = 4 (answer 610). (Limit domain)

Example #2: A ship is moving at a speed of 40 km/h parallel to a straight shoreline. The ship is 7 km from the shore and it passes a light house at noon.

Express the distance s between the light house and the ship as a function of d, the

distance the ship has traveled since noon. (answer s = f(d) = ) Express d as a function of t, the time elapsed since noon. d = g(t) (answer d = g(t) =

40t) Find f(g(t)) (answer f(g(t)) = .

Example #3: A store selling very expensive items will, on February 8, sell any item for $50 less than the listed price. On any day in February, the store will give a discount of 15% to any customer who can prove that he/she contributed to a local charity. Let x = listed price of an item, let P(x) be the price you pay for an item on Feb. 8, so P(x) = x – 50. Let D(x) is the price discounted at 15%, so D(x) = .85x. Discuss the two composite functions P(D(x)) and D(P(x)), which would be most advantageous for the customer, and which would be most advantageous for the store owner.Refer to section 1-2 Composition of Functions in the Pre-calculus Textbook. Focus attention to page 15, the domain and range of a composite function. For additional problems, use problems 1 – 33 on pages 17 – 19 in the textbook. You can also use the following web site for skill and drill.http://www.regentsprep.org/Regents/math/algtrig/ATP7/fogprac.htm (F.BF.1c, MP.1, MP.2, MP.4, MP.5, MP.6, L.2)

Give students teacher created assessment on T/L #1-6

7. Inverse (invertible) functions: Explain using inverses, since Algebra I you have used inverses to solve equations such as 2x = 6, divide both sides by 2 to get x = 3. Take a linear function y = 3x – 5 and solve for y and explain using the inverse operations in the steps. Then show the students switching the x and y and solving for y and replacing it with f -1(x). To check if it is an inverse, look at the relationship between the composite functions: f(g(x)) = g(f(x)).

Teachers choose one of the above story problems and have students find the inverse, for example in the ship problem have them find the inverse function and then give them a distance (s > 7) and have them find the time it takes to go that distance. When discussing this particular problem, it would be a great time to review the domain of the original function, values of t and their affect on f(g(t)). (F.BF.4b, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, L.2)

8. Provide examples of inverses that are not purely mathematical to further introduce the idea of inverses. (F.BF.4c, F.BF.4d, MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, L.1, L.2)

For example, given a function that names the capital of a state, f(Ohio) = Columbus, f(California) = Sacramento. The inverse would be to input the capital city and the state would be the output, f -1 (Columbus) = Ohio, etc. Now look at a function and inverse where the inverse is not a function (invertible function), such as a function that assigns a continent to a given country, f(USA) = North America. When examining the inverse f -

1(North America) = USA, f -1(North America) = Canada also. So the inverse is not a


http://www.regentsprep.org/Regents/math/algtrig/ATP7/fogprac.htm

TEACHING-LEARNING TEACHER NOTESfunction.

To extend this to mathematical functions consider f(x) = x2, the inverse is not a function. Have students decide which of the following functions have inverses that are functions: F(x) = x3 + 5x2, F(x) =4x, F(x) = 2|3x+4|, f(x) = 1/x. Have students create tables of values and discuss whether the inverses are functions or not by switching the x and y in the tables.

Have students then graph the functions and look at the vertical line test for the function and vertical line test for the inverse to determine if the inverse is a function or not. Examine symmetry with respect to y = x. (Review symmetry with respect to y = x if needed). Now that we have identified the ones that are not functions, discuss restricting the domain of the function so its inverse is a function – refer to the continent table first then progress to the other examples.

Add skill and drill practice problems on finding inverses and determining if they are functions or not and restricting the domain to make them functions (graph, table, algebraic). Refer to section 3-4 pages 152-157 in the textbook.

9. Graph both log and exponential functions and show they are symmetric with respect to y = x so they are inverses of each other. (F.BF.5, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, L.1, L.2)

10. Work on application problems involving logs and exponents (Refer to chapter 11 in the textbook):a) interest - worksheet #5 (attached on pages 10-11) b) decay and growth - worksheet #6 (attached on page 12) c) Extra problems for interest and decay - worksheet #7 (attached on pages 13-14)

TRADITIONAL ASSESSMENT TEACHER NOTES1. Paper-pencil test with M-C questions.

TEACHER CLASSROOM ASSESSMENT TEACHER NOTES1. 2 and 4 point questions2. Other projects, worksheets, etc..

AUTHENTIC ASSESSMENT TEACHER NOTES1. Students evaluate goals for the Unit.

2. Search for an exponential or log data set that is school appropriate, calculate a regression equation, find the inverse of it and prove it is the inverse by using composition of functions. Extrapolate to find a future data point and explain what it means. Also using the inverse, extrapolate when an event would occur in the future. (F.BF.1b, F.BF.4b, F.BF.4c, F.BF.5, MP.1, MP.2, MP.4, MP.5, MP.7, L.1, L.2)

ELEMENTS OF THE PROJECT 0 1 2

Data set – exponential or log Did not find a data set

Specified a data set – not exponential or logarithmic

Specified a data set that is either exponential or logarithmic

Regression equation Did not calculate a regression equation

Calculated a regression equation – does not represent the data or

Calculated a regression equation that represents the exponential or logarithmic data


http://www.teach-nology.com/worksheets/math/logarithms/log11.html

http://www.teach-nology.com/worksheets/math/logarithms/log10.html

equation not exponential or logarithmic

Inverse of function Did not calculate the inverse

Calculated inverse – contained an error.

Calculated the inverse correctly

Prove an inverse Did not prove an inverse

Proved an inverse – did not use composition of functions

Proved an inverse using composite functions

Find future data point and explained meaning

Did not find future data point

Found future data point – no explanation

Found future data point and explained meaning in relation to the problem

Using inverse, found data point and explained meaning

Did not find future data point with inverse

Found future data point using inverse – no explanation

Found future data point using inverse and explained meaning in relation to the problem

RUBRIC


Worksheet #1 Review laws of exponentsSimplify and write with positive exponents

1. 4a3b-2 * 2a2b5

2. (3a5b-3)2

3.

4.

5. -271/3

6. (161/4 a3/2 b2)* (3 a5/2 b)

_____________________________________________________________

Answers to worksheet #1:

1. 8a5b3 4.

2. 5. -3

3. 2a3b5 6. 6 a4 b3


Worksheet #2© This math worksheet is from www.teach-nology.comNatural Logarithm Operation WorksheetsSimplify and condense each problem.

1. ln 5 – ln 4 + ln 7

2. ln 9 – ln 3 - ln 2

3. ln 7 + ln 8 – ln 3 - ln 9

4. ln 12 + ln 4

5. 4ln 2 – 2ln 5 + ln 50

6. ln 5 – ln 5 + ln 7 + ln 7

Answers to worksheet #2

1. ln

2. ln

3. ln

4. ln 48

5. ln 32

6. ln 49


Worksheet #3© This math worksheet is from www.teach-nology.comSolving Natural and Common Logarithms EquationsSolve for the unknown (z) in each problem. Round to a whole number.

1. ln (2z + 4) = 2.3026

2. ln (9 + z) = 2.7726

3. ln (3z + 4) = 3.8286

4. 2 ln (z - 2) = 2.1972

5. ln (4z - 6) = 2.303

6. ln (2z - 4 +8) = 3.5264

7. log (4z – 10) - 1 = 1.64

8. log (z + 3)2 = 3.45

9. log (5z – 6)¼ = 0.125

________________________________________Solving Natural Logarithms EquationsAnswer Key1. 3 7. 112 2. 7 8. 503. 14 9. 24. 55. 46. 15


Worksheet #4

Solve each equation:

1. e2x = 24

2. ex+1 = 5

3. 4e5x = 32

4. 5e2x-1 + 10 = 35

5. 8e3x – 6 = 58

6. 5x = 10

7. 23x+1 = 5

8. 3 * 41-5x = 27

Answers to worksheet #4

1. 1.592. 0.613. 0.424. 1.35. 0.696. 1.437. 0.448. -0.12


Worksheet #5: Compound interest: A = P(1+ )nt, where A is the final amount of the investment, P is the principal, r is the annual interest rate, n is the number of times the interest is compounded yearly, t is the number of years. © This math worksheet is from www.teach-nology.comLogarithms Word Problems

1. Jake invested $1,900 in four year stock market scheme that pays three percent annually. What is the compound interest and amount?

2. Leslie has a bond for which he invested $2,900. The bond pays three percent annually for the last two years. What is the compound interest and amount?

3. Wilbur invested $6,500 in a five year CD that pays eight percent compounded half yearly. What is the compound interest and amount that will be in the bank after five years?

4. Andrew invested $3,200 in a seven year bond which pays four percent compounded quarterly. What is the compound interest and amount that Andrew will get after seven years?

5. Jim borrowed $2,000 from a bank a year ago with a rate of seven percent. What is the simple interest and the amount he has to pay to the bank?

6. Gary invested $7,800 in some three year scheme that pays eight percent. Calculate the simple interest and amount that he'll get.


7. Using simple interest, what will be the rate if amount is $5000, principal is $4678, and time is two years?

8. Jerry borrowed $120 from Tom nine months ago. Today, Jerry paid Tom back $137.What is the simple annual interest rate that Paul paid?

Logarithms Word ProblemsAnswer Key1. Interest = 238.47 Amount = 2138.472. Interest = 176.61 Amount = 3076.613. Interest = 3121.59 Amount = 9621.594. Interest = 1028.13 Amount = 4228.5. Interest = $140 Amount = $21406. Interest = $1872 Amount = $96727. Rate = 3.48. Rate = 18.9


Worksheet #6: Exponential growth or decay: N = N0(1+r)t, where N is the final amount after time t, N0 is the initial amount, r is the rate of growth (positive r) or decay (negative r) per time period, and t is the number of time periods.© This math worksheet is from www.teach-nology.comDecay and Practical EverydayLogarithms Word Problems

1. A material decays at a rate of 0.84% per year. How much of 420 grams of the material will be left in 7 years?

2. A material decays at a rate of 0.72% per year. How much material will be decayed after 14 years, if it weighs 210 grams today?

3. A material decays at a rate of 0.22% per year. How much of 98 grams of the material will decay in 6 years?

4. Mary pays a $712 premium for house loan. If the premium increases at an annual rate of 6% per year, how many years will it take for the premium to be $898.90?

_________________________________________Decay and Practical EverydayAnswer Key1. 395.30 grams2. 16.63 grams3. 1.3 grams4. 4 years


Worksheet #7© This math worksheet is from www.teach-nology.comMixed Logarithms Word Problems

1. Casey invested $1,225 in a certain bond that pays 19% interest compounded annually. How long will it take for Brad's investment to triple?

2. A material of 490 grams decays at a rate of 0.94% per year. How much material will be left after 9 years?

3. Elsie put $3,800 in a scheme that pays 9% interest compounded quarterly. How long will it take for Elsie's investment to double?

4. A material decays at a rate of 0.58% per year. How much of 240 grams of the material will have decayed in 7 years?

5. Mike's bond will be worth $15776.60 in ten years. The bond has an interest rate of 6% that is compounded half-yearly. What is the present value of the bond?

6. Mini put $9,870 in a three year CD that pays nine percent compounded annually. What is the compound interest and amount that will be in the bank after three years?

7. Mary's health bond will be worth $5713.81 in six years. If the interest rate is 6% that is compounded quarterly then what is the present value of the bond?


8. Gabriel invested $2,100 in a two year scheme that pays seven percent compounded quarterly. What is the compound interest and amount that will be in the bank after two years?

_____________________________________________________________Mixed Logarithms Word ProblemsAnswer Key1. 6 years and 3 months2. 445.8 grams3. 7 years, 9 months4. 9.5 grams5. Present worth = $8735.126. Interest = $2911.93 Amount = $12,781.937. Present worth = $3997.068. Interest = $312.65 Amount = $2412.65


youngstown city schools€¦ · web viewworksheet #2(attached on page 7) to reinforce properties...

Documents