Algebra 2 Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED

Transcription

1 Algebra Unit 8 (Chapter 7) CALCULATORS ARE NOT ALLOWED. Graph eponential functions. (Sections 7., 7.) Worksheet 6. Solve eponential growth and eponential decay problems. (Sections 7., 7.) Worksheet 8. Simplify arithmic epressions. (Section 7.) Page Worksheet 7 w. Common and natural s, inverse properties of and graph arithmic equations. Worksheet 9 w, b, 5. Apply the laws (properties) of s. (Section 7.5) Page 50 5 Worksheet Approimating arithmic values. Change of base theorem. Worksheet 6 7. Solve eponential equations with a common base. Worksheet 7-0 Page Solve arithmic equations (Section 7.6) Worksheet 8 9. Solve equations. (Section 7.6) Worksheet 9 0. Solve eponential equations without a common base. Worksheet 0 - Review Review Worksheet 90 Review Worksheet 5 Review Worksheet - - -

2 Algebra Unit 8 Worksheet CALCULATORS ARE NOT ALLOWED Simplify: Definition: The function defined by y = b is called an eponential function with base b Requirements: b > 0, b Characteristics of eponential functions: The basic graph of an eponential function looks like the following: Increasing An increasing eponential if they rise as they go from left to right. Decreasing A decreasing eponential if they drop as they go from left to right. Other characteristics: The -ais is a horizontal asymptote of the graph and the graphs contain the point (0,). In problems 6, complete the table of values and then graph on graph paper.. y =. y = 5. y y y = 6. y y = 5 y

3 Sketch the following graphs of the eponential functions and state if they are increasing or decreasing graphs. Be sure to label the intercepts. 7. y = 8. y = 9. y = 5 0. y Create a table of values in problems - and then graph on graph paper.. y =. y = 0. y = ( ) =. Eplain why the graphs of #- are not eponential functions. What in the equations is wrong? Answer the following multiple choice questions based on your knowledge of eponential functions and their graphs. Pay attention to increasing and decreasing equations. 5. If the equation of y = 5 is graphed, which of the following values of would produce a point closest to the -ais? a. 0 b. c. d If the equation of y = is graphed, which of the following values of would produce a point closest to the -ais? a. b. c. 5 d If the equation of y = is graphed, which of the following values of would produce a point closest to the -ais? a. 0 b. c. d Which multiple choice ordered pair represents the y-intercept for the function y =? a. (0,0) b. (0, ) c. (0, ) d. there is no y-intercept 9. Select the correct multiple choice response. The graph of y = 5 lies in which quadrants? a. Quadrants and b. Quadrants and c. Quadrants and - -

4 0. Select the correct multiple choice response. The graph of y = 0 contains which of these points? a. (0, 0) b. (0, 0) c. (0, ) d. (0, 0 ). Which multiple choice ordered pair represents the -intercept for the function y =? a. (0, 0) b. (0, ) c. (, 0) d. there is no -intercept. Use the graph of y = to answer the following multiple choice question. If the equation y = is graphed, which of the following values of would produce a point closest to the -ais? a. b. c. 5 d. 8. Given the epression n where > and n >, which multiple choice statement is true? a. the value of n = 0 b. the value of n > 0 c. the value of n < 0 d. the value of n =. Given the epression n where > and n = 0, which multiple choice statement is true? a. the value of n = 0 b. the value of n > 0 c. the value of n < 0 d. the value of n = 5. Given the equation y = n where 0 < < and n >, which multiple choice statement is true? a. y = 0 b. y > 0 c. y < 0 d. y = 6. Given the equation y = n where > and n < 0, which multiple choice statement is true? a. y = 0 b. y > 0 c. y < 0 d. y = - -

5 Algebra Unit 8 Worksheet CALCULATORS ARE NOT ALLOWED Many real world phenomena can be modeled by functions that describe how things grow or decay as time passes. Eamples of such phenomena include the studies of populations, bacteria, the AIDS virus, radioactive substances, electricity, temperatures and credit payments, to mention a few. Any quantity that grows or decays by a fied percent at regular intervals is said to possess eponential growth or eponential decay. Such a situation is called Eponential Decay. Such a situation is called Eponential Growth. The time required for a substance to decay and fall to one half of its initial value is called the half-life. Radio-isotopes of different elements have different half-lives. Some people are frightened of certain medical tests because the tests involve the injection of radioactive materials. Doctors use isotopes whose radiation is etremely low-energy, so the danger of mutation is very low. The half-life is long enough that the doctors have time to take pictures, but not so long as to pose health problems. They use elements that are not readily absorbed by the body but are voided or flushed long before they get a change to decay within your body

6 For the following word problems we will be using the eponential equation y = A ( b ) t h. Where A is the initial amount b is the amount of growth (or decay) that occurs in h time t is time. Technetium-99m is one of the most commonly used radioisotopes for medical purposes. It has a half life of 6 hours. If 0.5 cc s (which is less than a teaspoon) of Technetium-99m is injected for a scan of a gallbladder, how much radioactive material will remain after hours? Use the formula y = A t 6 where A = the number of cc s present initially t = time in hours. When a plant or animal dies, it stops acquiring Carbon- from the atmosphere. Carbon- decays over time with a half-life of 570 years. How much of a 0mg sample will remain after,60 years? Use the formula N = N 0 t h where N 0 is the initial amount N = the amount remaining t = time in years h = half life. One certain element has a half-life of 600 years. If 00 grams were present originally, how many grams will remain after 00 years?. The radioactive gas radon has a half-life of days. How much of an 80 gram sample will remain after 9 days? 5. The radioactive gas radon has a half-life of approimately days. About how much of a 00 gram sample will remain after week? 6. The population of a certain country doubles in size every 60 years. The population is now million people. Find its size in 80 years. y = A () 60 t A = initial population t = time elapsed in years - 6 -

7 7. Bacteria populations tend to have eponential growth rather than decay. Suppose a certain bacteria population doubles in size every hours. If you start with 00 bacteria how many will there be in 8 hours? 8. A certain population of bacteria doubles every weeks. The number of bacteria now is only 0. How many will there be in 5 weeks? 9. A culture of yeast doubles in size every 0 minutes. The size of the culture is now 70. Find its size in hour (remember to convert hour to minutes.) 0. The growth of a town doubles every year. If there are 6,000 people after years, find the initial population.. The number of people with a flu virus is growing eponentially with time as shown in the table below. Flu Virus Growth Day Number of People Which multiple choice equation epresses the number of bacteria, N, present at any time,? a. N = 00 b. N = 00 + c. N = 800 d. N = 00. In the early years of the century the national debt was growing eponentially with time as shown in the table below. National Debt Year Debt 0 0,000 60,000 0,000 Which multiple choice formula epresses the debt, y, at any time t? a. y = 0,000 t b. y = 0000 t c. y = 0 t d. y = 0,000 + t. An epidemic of bubonic plague grew eponentially by the formula A = A 0 t where A 0 = original amount infected t = time passed in weeks If 5,000 people were infected after 8 weeks, find the original amount that were infected

8 Use estimation for the following multiple choice questions: Choose the best multiple choice response for the following:. 9 a.. b..9 c..8 d ( ) ( ) a. 6.9 b..9 c. 67. d A radioactive element decays over time according to the equation: y = A If 000 grams were present initially, how may grams will remain after 650 years? a. b. c. d Boogonium decays using the formula: A = I t h The half life of Boogonium is hours. How much of a gram sample will remain after 6 hours. Choose the best multiple choice response. a. 0. b.. c. 8.5 d Geekonium-5 decays using the formula: A = I t h The half life of Geekonium-5 is years. Find how much of a 60 gram sample remains after 8 years. t 00 Unit 8 Worksheet Determine the eponent needed to change the left number into the right number. You may use positive, negative, zero, and fractional eponents. Guess and Check: ½. / /

9 Logarithms (or s) are used to find the eponents to help us solve eponential equations. Structure of a arithm: y = b Eample: Simplify 8=? b is the base y is the value is the eponent on b to yield y = (because = 8) Simplify # Logarithms with a base 0 are called common arithms. The base of 0 is implied and not shown. For eample, 000 is equivalent to Simplify: (Remember, when no base is given it is assumed to be base 0)

10 Unit 8 Worksheet Log Rules : (b and y must be positive numbers, b ) b y = b = y b b = y b b = y b = 0 π. e.78 Remember, if no base is shown assume it is base 0, the common. y = 0 y = 0 = y If base e is used it is called a natural. Instead of writing we use ln ln y e = y = e y = (Remember, e is just an irrational number. It is approimately.78; see Page 9 in your tetbook) Restrictions: You can t take 0 or (of a negative number) With bases, you can t do 0 base or base or negative base Verify the by rewriting the equation into eponential form.. = 5. 9 =. 7 7 =. 8 = Rewrite the equation in arithmic form. 5. = = = = 8 Simplify:

11 Write each equation in eponential form.. ln 8 =.08. ln 00 = π =. 000 = 5. ln 097 = 7 To graph a equation:. First rewrite it in eponential form. Make a table of values. Look at the equation and see which letter ( or y) is the eponent and put the numbers,, 0,, in that column.. Plot the points and connect with a curve Graph #6-9 on graph paper. Be sure to show the table of values and the eponential equation. 6. y = 7. y = 5 8. y = 9. Graph y = and y = on the same grid. Choose the correct multiple choice. 0. Which is equivalent to 6 =? a. = 6 b. 6. Which is equivalent to m n = p? = c. 6 = d. 6 = a. m n = p b. m p = n c. n p = m d. p n = m. Which is equivalent to k = w? a. 0 w = k b. w = k c. k w = 0 d. 0 k = w. Given: y = 5 which statement is true? a. y > 0 for all values of b. y > 0 for all values of c. y < 0 for all values of d. y < 0 for all values of. When is the following statement true? 7 7 = a. for all values of b. for some values of c. for no values of d. can t determine - -

12 5. In the equation y = z which statement is true about the value of z? a. z must always be positive b. z can never equal 0 c. z can never equal d. there are no restrictions on z 6. When is the equation 6 6 y = y? a. for all values of y b. for no values of y c. for some values of y d. cannot determine 7. Which epression is equivalent to ln = y? a. 0 y = b. e y = c. y = e d. e y = 8. Which epression is equivalent to 6 6? a. b. 6 c. d Which epression is equivalent to 000? a. 000 b. c. 0 d. - -

13 Unit 8 Worksheet 5 On pg. 507 in our tet are the Laws of Logarithms. Multiplication Property: b MN = b M + b N. Quotient Property: b M N = b M b N n. Power to a Power Property: M b = If you are given: 0 =.60 and 0 6 =.778, use the Laws and the given to find the following. Justify each step with the properties listed above or basic operations property. Eample ( 6) Factors of Multiplication Property Substitution Property.80 Addition n b M (hint: = ) 8. 0( ) 6 and 0 6 we know 00 = and 000 = Even though we were only given In the preceding problems we had to work with decimal values. The following problems involve the same laws of arithms, but we will use variables instead of decimals. Given: 9 = c and 0 = d Find the following in terms of c and d =. 8 =. ( ). 0 = 5. ( ) 9 = 9 = 6. ( ) = 9. ( 9) = 0. You were given the 9 and 0, but you also know =, use this to find 8 = - -

14 Select the correct multiple choice:. y = a) y b) + y c) + y d) + y. y = a) ( + y) b) ( y) c) + y d) none of these. y = a) y b) y c) both a and b d) neither a or b. 00 = a) b) 6 c) 8 d) 6 5. = a. + b. c. + d. 6. = a. ( ) b. c. d y + z = a. ( + y + z) b. ( y z) c. y z 8. ( w ) = a. w b. w c. w d. w 9. Which student solved for correctly in the following problem? = Alice Bob Carl David = = = = = = = = = = = 0 = 0 = = ± = 0000 = 0000 = 00 = ±

15 0. Which student solved for correctly in the following problem? + = 6 Astro Bella + = 6 + = = = 6 9 = 6 (9 + ) = 6 9 = = 6 = = 7 Chu Domingo + = 6 + = 6 ( + ) = 6 ( + ) = 6 = 6 = 6 = 6 () = 6 = 6 9 = 6 = = = = Unit 8 Worksheet 6 A. If we write 0 in eponential form we get 0 = We are going to have to = 8 approimate the value of this. We know = 0 = 6 So the eponent,, will be between the consecutive integers and. B. 5 becomes = 5 Between what consecutive integers will lie? = 9 = 5 = 7 So is between and. Would it be closer to or closer to? Determine which two integers the following arithms lie between: You can convert all arithm problems to equivalent arithms with base 0 or e. Below is the formula to convert arithms to any base. Change of Base

16 c a is currently in base c. To change it, write it as a fraction c a = a c You ll notice that no base was given. You can use any base. For eample: c a = Change of Base Formula a c = 6 a c or a c or 8 a c 6 8 c a = a b b c (where b can be any positive base ) Since most calculators only work in base 0 or base e, it is best to change to one of them. c a = a 0 ln a 0 c or ln c Rewrite the following using the change of base formula. Change into the indicated base to base 7. 9 to base 6 8. to base to base e You can use the change of base formula in reverse. If you are given a b c b you can condense it to a single by dropping the base b. a b b c = c a Epress the following as a single : - 6 -

17 Epress the following as a single. Then simplify the final answer = ln ln a. 5 7 b. 7 5 c. 7 5 d =. ln ln 7 5 a. 0 8 b. 0 8 c. 0 8 d = 7 a b. 8 6 c. d. Solve for using common bases.. =. 7 Algebra Unit 8 Worksheet =. = 8. 7 = 5. = = 9 ( + 5) = = = = =. 6 = 6 6 Solve for using inverse properties of eponents.. = 5. = = 6. = = = ( + 5) = 0 0. ( ) = 0-7 -

18 Unit 8 Worksheet 8 Solve for. Some problems may have no solution.. =. =. 5 =. ( ) = 5. = = 7. = 8. 8 = 9. 8 = 0. 6 =. 6 6 =. =. 7 =. 7 ( 9) = 5. ( 9) = 6. 6 = = = 9. 9 = 0. 8 =. 6 =. (7 ) =. 0 5 =. 8 = 5. 5 (5 ) = 6. ( 5 ) = 7. 7 = = = ln 9 = ln. 7 = ( + ) = 5 ( + ). 8 = = 8-8 -

19 Algebra Unit 8 Worksheet 9 Solve for using properties of s. On problems involving π or e leave answers in terms of π or e. Do not approimate. Some problems will have no solution.. 7 = = ( + ) = ( 5) = 6 5. ln ( + 5) ln ( 5) = ln 8 6. = = = = 0. 5 = =. ln = ln 8. π =. π5 + π = = ( 5) = 7. = 8. ln = 9. ln + ln 5 = 0. ln ln 6 =. ( ) =. + ( 6) =. + =. ln 7 + ln = = = 7. 5 ( 7) = 0 8. ln ( 9) = 9. Identify which step has the error in the solution of 7 = Step : 7 = 7 ( 50) Step : 7 = 7 00 Step : 7 = 7 00 Step : 7 = 7 50 Step 5: =

20 0. Which line has an error in it? = =. 6 = = = 6 5. =. What multiple choice helps when solving =? a. = 6 b. = 6 c. = 5 d. =. What multiple choice helps when solving = 5 a. + y = ( + y) b. + y = (y) c. p = p d. y = y. What multiple choice helps when solving ln = a. ln = ln e b. e.78 c. = d. 0 = - 0 -

21 If we are given = Unit 8 Worksheet 0 CALCULATORS ARE NOT ALLOWED 5, how would we solve for the eponent,? We use arithms to help us solve these eponential functions. Equation: = 5 = 5 5 = (our calculator could give us a decimal approimation, but for now this is how we write our answers) Solve the following problems for by introducing s. Leave answers in form.. 7 =. 5 = 0. 0 = 9. 8 = = 6. e = Choose the correct multiple choice response: 7. 7 = a. = b. = c. = 7 d. = 8. If = 5 which is true about? a. < 0 b. 0 < < c. < < d. > 9. 0 = 00 a. = 00 b. = 00 0 c. = 0 d. = 0 0. e = a. = b. = ln c. = ln e d. =. + = a. = b. = 6 c. = d. = 6 - -

22 . Which step has the error: ln 8 + ln = 5 Step ln 8 = 5 Step 8 = 0 5 Step 8 = 00,000 Step = 00, Which step has the error: 7 + = 9 Step = 7 Step + = 7 9 Step = 7 9 Step = 7 8 Algebra Unit 8 Review CALCULATORS ARE NOT ALLOWED Simplify: Write the following in arithmic form. 5. = = e = b a = c Write the following in eponential form = 0. 5 = =. ln 8 = 5. 7 = 0. π = Simplify. Some problems will have no answer ln. 8. e ln ( e ) 0. ln () 8 ( )

23 ln ( e ) Solve for. On problems involving π or e leave answers in terms of π or e. (Do not approimate.) Some problems will have no solution. Some problems will have answers in terms.. =. =. = 6. 6 = 5. 5 = = 7. = = 7 9. (7 ) = 0 0. e =. 9 = 7. e + = 0. 5 =. = 5. 5 = 6. ln = ( 5) = 8. π π = 9. ln e = 50. ln = = = ( + ) 7 = 7 5. ln () + ln () = ln ( + ) ( ) = = 6 75 Epress as a single and simplify, if possible ln8 ln Given: =.00 6 =.778 Find the following: (this is equivalent to )

24 Given: = k 5 = f Find the following: Graph the following: 77. y = y = 79. y = Answer individual questions: 80. Between what consecutive integers does 0 lie? 8. If the equation y = is graphed, which of the following multiple choice values of would produce a point closest to the -ais? a. b. 0 c. d. 8. A radioactive substance decays by the given formula. How much of a 60 gram sample will remain after 6 hours? y = A t A = initial amount t = time in hours 8. A radioactive element decays over time as shown in the table below. Which multiple choice equation epresses the amount of grams, y, present at a. y = h g b. y = 00 hour, h? hour grams h c. y = h g d. y = Given the equation y =, which multiple choice statement is valid? a. < 0 b. < 0 c. = 0 d. > Which multiple choice is equivalent to 0 5 a. b. 5 c. 86. Which multiple choice is equivalent to 6? a. b d. 00 c d. ( 6 )( 6 ) - -

25 87. Which multiple choice is the solution to the equation 9 = 5? a. = 5 9 b. = 5 c. = 5 d. = Given the epression n where > and n >, which multiple choice statement is true? a. the value of n = 0 b. the value of n > 0 c. the value of n < 0 d. the value of n = 89. Given the epression n where > and n =0, which multiple choice statement is true? a. the value of n = 0 b. the value of n > 0 c. the value of n < 0 d. the value of n = 90. Given the equation y = n where 0 < < and n >, which multiple choice statement is true? a. y = 0 b. 0 < y < c. y < 0 d. y = Algebra Unit 8 Review CALCULATORS ARE NOT ALLOWED Choose the correct multiple choice response in #.. Write 7 = in arithmic form. a) 7 = b) = 7 c) 7 = d) 7 =. Write = in eponential form a) = 0 b) = c) = d) 0 =. Evaluate 6 a) b) c) d) e). Solve for : 9 = a) b).5 c), d) 8 e) 5. Solve for : 5 = a) 5 b) 5 c) d) e) Evaluate: 5 5 a) 5 b) 6 c) 5 d) 6 e) none of these Evaluate: 7 a) 7 b) c) d) 9 e) none of these 8. Solve: ( 8) = a) b) c) d) e) none of these 5 9. Solve: y = 5 a) 5 b) c) 75 d) e) none of these 0. Solve: = a) b) c) 6 d) e) none of these. Solve: ( m ) + ( m ) = a) b) 5 c) 9 d), 5 e) none of these. Solve: () + ( + ) = a) b) c) - d), e),. Solve: 9 = a) b) - c), - d) e) none of these - 5 -

26 . Given: = c and 7 = d, Find: 56 a) c + d b) c d c) c + d d) c + d e) none of these 5. Solve for : ( ) = a) b) c) 8 d) e) none of these 6. Solve for : 5 0 = a) 0 b) c) 5 d) e) none of these 7. Solve for : = 5 a) b) c) d) e) none of these 8. If 7 = n, find ( ) a) b) n 7 n c) n d) n e) none of these 9. If = c and = d, find 6 a) c+ d b) c + d c) cd d) c + d e) cd f) none of these 0. Which of the following is true about the graph of y =? a) it passes through (0,) b) it lies in quadrants and only c) it is a decreasing graph d) the value of will never be 0. Solve for : = 7 a) 6 b) 6 c) 56 d). Find the value of : = 5 a) 50 b) 5 c) 5 d) 65 True or False. 0 = = = ln e = = = Simplify: Solve for :. =. 6 ( + 5) + 6 = = = 6. (7 ) = 7. = 8 8. = 8 9. ln = 0. ( 9) = - 6 -

27 . Graph: y =. Graph: y = 5. Epress as a single : ln 7 + ln. Epress as a single and simplify: If 5 = k and 5 = m find 5 0 in terms of k and m 6. Given the equation y = n where > and n < 0, which multiple choice statement is true? a) y = 0 b) y > 0 c) y < 0 d) y = 7. If the equation y = is graphed, which value of would produce a point closest to the ais? a) b) 5 c) d) 8. If the equation, y = is graphed, which value of would produce a point closest to the -ais? a) 7 b) 0 c) d) If the equation, y = is graphed, which of the following values of would produce a point 7 closest to the ais? a) b) c) d) If the equation, = y is graphed, which of the following values of would produce a point farthest from the ais? a) 8 b) c) d) 9 Simplify Unit 8 Review # n. Given the equation y = where > and n > 0, which statement is true? a) y = 0 b) y < 0 c) y > d) 0 < y < e) y is undefined n. Given the equation y = where 0< < and n > 0, which statement is true? a) y = 0 b) y < 0 c) y > d) 0 < y < e) y is undefined. Given the equation y n = where 0 < < and n < 0, which statement is true? - 7 -

28 a) y = 0 b) y < 0 c) y > d) 0 < y < e) y is undefined. Bacteria are growing eponentially with time as shown in the table below. Write the equation that epresses the number of bacteria, y, present at any time, t? Bacteria Growth Hour Bacteria Bacteria are decaying eponentially with time as shown in the table below. Write the equation that epresses the number of bacteria, y, present at any time, t? Bacteria Growth Hour Bacteria Simplify the following: 7. ( ) ( 9) Approimate the following: