Chapter 8 Prerequisite Skills

Transcription

1 Chapter 8 Prerequisite Skills BLM 8. How are 9 and 7 the same? How are they different?. Between which two consecutive whole numbers does the value of each root fall? Which number is it closer to? a) 8 c) 9 d) 00. Identify two rational numbers with square roots between 8 and 9.. Identify the base and the eponent in each of the following powers. Evaluate each power where possible. a) () c) 7 d) e) f) g).78.. Calculate. a) c) 96 d) 7 e) 96 f) 9 6. Write each epression as powers without parentheses. Then, evaluate each epression. a) ( ) (7 ) c) 6 d) [() ] 7. Determine the value of each epression. a) 7 ( ) ( ) () c) () 6 d) (7 ) 8. For each table, plot the ordered pairs (, y) and the ordered pairs (y, ). State the domain of the function and its inverse. a) y y Sketch the inverse of each graph of a relation. a) 0. Determine algebraically the equation of the inverse of each function. a) f () f () c) f () = d) f () e) f (). f) f () ( 6). For each of following functions, determine the equation for the inverse, f () sketch the graph f () and f () state the domain and range of f () and f () a) f () f () c) f () = ( 6) d) f (), 0 e) f (), 0 f) f () ( ), Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

2 Section 8. Etra Practice BLM 8. Use the definition of a arithm to evaluate each epression. a) c) 8 d) 8 e) 7 f) g) 0.0 h) 6. Epress in arithmic form. a) 6 c) 0. d) m n. Epress in eponential form. a) 6 8 c) d) 6 ( ) y. Determine the value of. a) c) 8 d). a) Sketch the graph of the eponential function y. On the same grid, sketch the graph of the inverse of y. c) Eplain the relationship between the characteristics of the two functions. 6. a) State the equation of the inverse of f( ). Sketch the graph of the inverse. c) Identify the domain, range, and intercepts of the inverse graph. d) Determine the equations of any asymptotes. 7. Identify the following characteristics of the inverse graph of each function. i) the domain and range ii) the -intercept, if it eists iii) the y-intercept, if it eists iv) the equation of the asymptote a) 8. Without using technoy, estimate the value of each arithm to one decimal place. a) 60 0 c) 80 d) 9. a) Determine the -intercept of y ( ). Determine the y-intercept of y The point 6, is on the graph of the arithmic function f () c. The point (k, 6) is on the graph of the inverse, y f (). Determine the value of k. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

3 Section 8. Etra Practice BLM 8. Describe how the graph of each arithmic function can be obtained from the graph of y. a) y ( 8) y () c) y ( 0) 9. a) Sketch the graph of y. Then, apply, in order, the following transformations. Stretch horizontally by a factor of about the y-ais. Translate units to the right. Write the equation of the final transformed image.. a) Sketch the graph of y 6. Then, apply, in order, the following transformations. Reflect in the -ais. Translate vertically units down. Write the equation of the final transformed image.. Sketch the graph of each function. a) y ( ) 7 y ( ( )) c) y (). Identify the following characteristics of the graph of each function. i) the equation of the asymptote ii) the domain and range iii) the y-intercept, to one decimal place if necessary iv) the -intercept, to one decimal place if necessary a) y () y (( )) c) y 7 ( ) d) y ( 0) 6. In each graph, the solid curve is a stretch and/or reflection of the dashed curve. Write the equation of each solid graph. a) c) d) Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

4 BLM 8 (continued) 7. Describe, in order, a series of transformations that could be applied to y to obtain the graph of each function. a) y (( )) 7 y 0. (( )) 8. The graph of y has been transformed to y a (b( h) k. Determine the values of a, b, h, and k for each set of transformations. Write the equation of the transformed function. a) a reflection in the y-ais and a translation units right and units down a vertical stretch by a factor of about the -ais and a horizontal stretch about the y-ais by a factor of c) a vertical stretch about the -ais by a factor of, a horizontal stretch about 9. Describe how the graph of each arithmic function can be obtained from the graph of y 7. a) y ( ) 7 y 0. ( ) c) (y 7) ( ) 0. a) Only a horizontal translation has been applied to the graph of y so that the graph of the transformed image passes through the point (6, ). Determine the equation of the transformed image. A vertical stretch is applied to the graph of y so that the graph of the transformed image passes through the point (, ). Determine the equation of the transformed image. the y-ais by a factor of, a reflection in the -ais, and a translation of 7 units left and units up Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

5 Section 8. Etra Practice BLM 8. Write each epression in terms of the individual arithms of, y, and z. a) y 7 z ( yz ) c) ( yz ) d) y z. Use the laws of arithms to simplify and evaluate each epression. a) c) d) 9. Write each epression as a single arithm in simplest form. a) y 6 6 y 6 z y c) d) y. Evaluate each of the following. a) If, determine the value of. Determine the value of n ab if n a and n b. c) If c, evaluate 0c. d) If a and a y, evaluate a y.. Simplify. a) a a 8 a 6. If 9 k, write an algebraic epression in terms of k for each of the following. a) 9 c) (8 ) 9 d) 7. Write each epression as a single arithm in simplest form. State any restrictions on the variable. a) 8. In chemistry, the ph scale measures the acidity (07) or alkalinity (7) of a solution. It is a arithmic scale in base 0. If neutral water has a ph of 7, what is the ph of a solution that is times more alkaline than water? 9. If bleach has a ph of, how many times more alkaline is it than blood, which has a ph of 8? 0. An earthquake off the coast of Alaska measured 6. on the Richter scale. Another earthquake near Japan was 0 times worse. What was the Richter scale reading for the earthquake near Japan? Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

6 Section 8. Etra Practice BLM 8. Solve. a) ( ) ( ) ( ) 6 c) ( ) (6 ). Solve. a) ( 7) ( ) c) ( ) ( ). Solve. Round your answers to two decimal places. a) 9 60 c). Determine the value of. Round your answers to two decimal places. a) 7 8 c) ( ). The following shows how two students chose to solve. Nicole s work: - = = = = =96 Joseph s work: - = = = = 96 = Which method of solving do you prefer and why? 6. The following shows how Samuel attempted 00 to solve the equation. 00 = 00 = = Identify, describe, and correct Samuel s errors. 7. Solve and check each solution. Round to two decimal places when necessary. a) ( ) ( ) 0 ( ) ( ) c) ( ) d) 9 ( ) 9 ( ) 8. The compound interest formula is A P( i) n, where A is the future amount, P is the present amount or principal, i is the interest rate per compounding period epressed as a decimal, and n is the number of compounding periods. a) Livia inherits $000 and invests in a guaranteed investment certificate (GIC) that earns 6% interest per year, compounded semi-annually. How long will it take for the GIC to be worth $0 000? How long will it take for money invested at.% interest per year, compounded semi-annually, to triple in value? 9. The population of a town changes by an eponential growth factor, b, every years. If a population of 0 grows to 7000 in years, what is the value of b? Round your answer to two decimal places. 0. Light passing through murky water loses 0% of its intensity for every metre of water depth. At what depth will the light intensity be half of what it is at the surface? Round your answer to two decimal places. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

7 Chapter 8 Study Guide BLM 8 6 This study guide is based on questions from the Chapter 8 Practice Test in the student resource. Question I can Help Needed Refer to # sketch and determine the characteristics of the graph of y = c, c > 0, c # epress a arithmic function as an eponential function and vice versa # eplain the effects of the parameters a, b, h, and k in y = a c (b( h)) + k on the graph of y = c, where c > # determine an equivalent form of a arithmic epression using the laws of arithms # determine an equivalent form of a arithmic epression using the laws of arithms #6 solve a problem by applying the laws of arithms to arithmic scales some none some none some none some none some none some none #7 solve a arithmic equation and verify the solution some none #8 evaluate arithms using a variety of methods some none #9 eplain the effects of the parameters a, b, h, and k in y = a c (b( h)) + k on the graph of y = c, where c > #0 sketch the graph of a arithmic function by applying a set of transformations to the graph of y = c, where c >, and state the characteristics of the graph some none some none # solve a arithmic equation and verify the solution some none # solve an eponential equation in which the bases are not powers of one another # solve a problem that involves the application of eponential equations to loans, mortgages, and investments # solve a problem by applying the laws of arithms to arithmic scales # solve a problem by applying the laws of arithms to arithmic scales some none some none some none some none #6 solve a problem involving eponential growth or decay some none #7 solve a problem by modelling a situation with an eponential or arithmic equation some none Section 8. Eample Section 8. Link the Ideas Section 8. Eamples, Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eamples, Section 8. Eamples, Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eample Section 8. Eample Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

8 Chapter 8 Test Multiple Choice For # to 6, select the best answer.. The graph of f () b, b >, is translated such that the equation of the new graph is epressed as y f ( ). The domain of the new function is A { > 0, R} B { >, R} C { >, R} D { >, R}. The -intercept of the function f () is A B 0 C D. The equation y can also be written as A y y B C y D y. The range of the inverse function, f, of f (), is A { y y > 0, y R} B { y y < 0, y R} C { y y 0, y R} D { y y R}. A graph of the function y is transformed. The image of the point (, ) is (6, ). The equation of the transformed function is A y ( ) B y ( ) C y ( ) D y ( ) 6. If 7 y, then 9 equals y y A B C y D y Short Answer 7. If, epress in terms of. BLM Determine the value of algebraically. a) 6 c) d) ( ) e) ( 6) 9. Solve for. a) ( ) ( ) ( ) ( 7) ( ) ( ) c) ( ) 0. The point (6, ) lies on the graph of y b. Determine the value of b to the nearest tenth. Etended Response. Solve the equation 0, graphically and algebraically. Round your answer to the nearest hundredth.. Given f () and g() 9. a) Describe the transformation of f () required to obtain g() as a stretch. Describe the transformation of f () required to obtain g() as a translation. c) Determine the -intercept of f (). How can the -intercept of g() be determined using your answer to parts a) or?. Eplain how the graph of ( ) y can be generated by transforming the graph of y.. Identify the following characteristics of the graph of the function y ( ) a) the equation of the asymptote the domain and range c) the -intercept and the y-intercept. An investment of $000 pays interest at a rate of.% per year. Determine the number of months required for the investment to grow to at least $000 if interest is compounded monthly. 6. Radioactive iodine- has a half-life of 8. days. How long does it take for the level of radiation to reduce to % of the original level? Epress your answer to the nearest tenth. Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

9 Chapter 8 BLM Answers BLM 8 8 BLM 8 Prerequisite Skills. Each is a root of a negative number. However, only 7 can be evaluated because 9 is not a real number.. a) and ; closer to and ; closer to c) and ; closer to d) and ; closer to. Eample: any rational number between 6 and 8. a) base: ; eponent: ; 8 base: ; eponent: ; () 0 c) base: ; eponent: 7 d) base: ; eponent: e) base: ; eponent: ; = f) base: ; eponent: ;.7 g) base:.78; eponent:.; a) 6 c) d) e) f) 7 6. a) () c) d) a) 8 c) d) 8. a) domain: { =,,, } 9. a) domain: { =,, 0,, } 0. a) f f c) f d) e) f 0.( ) f). a) f f f 6 domain: { =,, 0,, } f () domain: { R}; range: { y y R} f () domain: { R}; range: { y y R} f domain: { = 6,,,, } Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

10 BLM 8 8 (continued) f () domain: {, R}; range: { y y 0, y R} f) f f () domain: { R}; range: { y y R} f () domain: { R}; range: { y y R} c) f () f () domain: {, R}; range: { y y 0, y R} f () domain: { 0, R}; range: { y y, y R} BLM 8 Section 8. Etra Practice. a) c) d) e) 0 f). a) 6 f () domain: { R}; range: { y y R} f () domain: { R}; range: { y y R} d) f g) h) c) 0. d) (n ) m. a) 6 8 c) d) 6 y. a) 6 c) d) 8. a), f () domain: { 0, R}; range: { y y, y R} f () domain: {, R}; range: { y y 0, y R} e) f c) Eample: They are reflections of each other over the line y. Each point on the graph of one function (, y) appears as the point (y, ) on the other graph. 6. a) y f () domain: { 0, R}; range: { y y, y R} Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

11 BLM 8 8 (continued) c) domain: { 0, R}; range: { y y R}; -intercept: (, 0); y-intercept: none d) vertical asymptote at = 0 7. a) domain: { 0, R}; range: { y y R}; -intercept: (, 0); y-intercept: none; vertical asymptote at = 0 domain: { 0, R}; range: { y y R}; -intercept: (, 0); y-intercept: none; vertical asymptote at = 0 8. a).9. c).7 d). 9. a) (, 0) no y-intercept 0. k 6. a) BLM 8 Section 8. Etra Practice. a) translation horizontally 8 units left and vertically unit down reflection in the y-ais, stretch horizontally about the y-ais by a factor of c) reflection in the -ais, stretch vertically about the c) -ais by a factor of, translation horizontally 0 units right and vertically 9 units up. a) y. a) y 6. a) equation of asymptote: 0; domain: { 0, R}; range: { y y R}; y-intercept: none; -intercept: (, 0) equation of asymptote: ; domain: {, R}; range: { y y R}; y-intercept: none; -intercept: (., 0) c) equation of asymptote: ; domain: {, R}; range: { y y R}; y-intercept: (0,.); -intercept: (., 0) d) equation of asymptote: 0; domain: { 0, R}; range:{ y y R}; y-intercept: none; -intercept: (, 0) 6. a) y or y = y c) y () d) y = 7. a) a vertical stretch about the -ais by a factor of, a horizontal stretch about the y-ais by a factor of, a reflection in the -ais, and a translation units right and 7 units up a vertical stretch about the -ais by a factor of 0., a reflection in the y-ais, and a translation unit left and units down Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

12 BLM 8 8 (continued) 8. a) a ; b ; h ; k ; y (( )) a ; b 0.; h 0; k 0; y (0. ) c) a ; b ; h 7; k ; y 7 9. a) a vertical stretch about the -ais by a factor of, a horizontal stretch about the y-ais by a factor of, a reflection in the y-ais, and translation units right and 7 units down a vertical stretch about the -ais by a factor of 0., a reflection in the y-ais, and translation units right and units up c) a vertical stretch about the -ais by a factor of and translation unit left and 7 units up 0. a) y ( 0) y 9.0 BLM 8 Section 8. Etra Practice. a) 7 7 y7 z y z y z c) d) y z. a) 8 8 c).. d) 0. a) 6 y yz c) y 00 d) y. a) c) 7 d). a) 6 6. a) k k c) k d) 0.k 7. a), 0, times more BLM 8 Section 8. Etra Practice. a) no solution 9 c). a) 8 c). a).79.0 c).6. a).76.8 c) Eample: If Nicole's work is preferred it is because it uses the definition of arithm to convert into. Once this is done, the arithm can be dropped from both sides of the equation. If Joseph's work is preferred, it is because it converts the arithmic equation into an eponential function. 6. Eample: Samuel s error occurs in his first calculation: 00 divided by does not equal 00. To solve the equation correctly, Samuel should first calculate the of 00 and then divide this value by the of a).9 8 c) no solution d) 6 8. a). compounding periods, so.7 years 6. compounding periods, so.7 years 9. b m BLM 8 7 Chapter 8 Test. B. A. D. A. A 6. A a) c) d), e) 6 9. a). no solution c) a) horizontal stretch by a factor of about the y-ais 9 vertical translation units up c) -intercept of f () is ; the -intercept of g() is 9, since g() is a result of a horizontal stretch by a factor of 9. vertical stretch by a factor of a horizontal stretch by a factor of about the -ais, about the y-ais, a horizontal translation units right, and a vertical translation unit up Copyright 0, McGraw-Hill Ryerson Limited, ISBN:

13 BLM 8 8 (continued). a) domain: {, R}; range: 7 { y y R} c), y 8. 0 months 6..8 days BLM U Unit Test. A. A. D. B. C 6. D 7. a, k a) y (). (0.8, 0.8) and (, ); Eample: The two functions are inverses of each other. The points of intersection lie on the line y, the line of reflection.. a) 6 7 c). a) y, 0 y ( ), c) y 0.( ), {, R} 6. a) P(t) t.7% c) 0 7. a) moles per litre. 8. a) A t domain: {t t 0, t R}; range: {A A 00, A R}; no -intercept; y-intercept 00 c) years domain: { R}; range: { y y 0, y R}; no -intercept; y-intercept Copyright 0, McGraw-Hill Ryerson Limited, ISBN: