Suggested Problems for Math 22 Note: This file will grow as the semester evolves and more sections are added. CCA = Contemporary College Algebra, SIA = Shaum s Intermediate Algebra SIA(.) Rational Epressions pages 80-8. On Pages 94-95 Problems.,.,.4 Reduce the following rational epressios to lowest terms. () 72 08 (2) 282 y 6 2 5 () 8y + 6 (4) a 2 b 2 a 2 + 2ab + b 2 (5) 2 25 2 + 4 5 (6) 2 + 2 5 2 4 + (7) 4a b 2ab 6a b 2 (8) 82 +6y 2 + 4y (9) y 4 2 y 2 + 2 y y 2 y 4 + 2y 5 SIA(.2) Products and Quotients of Rational Epresions Pages 84-86. On Pages 95-96 Problems.5,.6,.7 Perform the indicated operations and simplify. () 7 0 8 (4) 7 + 0 7 + (2) 7 0 0 (5) + 7 (+ ) () 0 2 00 (6) + 2 9 (7) 22 y 5w 2 25w y 2 (8) 2 9 y 2 4 y 2 + 2y (9) 2 4 5 ( 9) 2 2 6 2 + 4 + 4 2 6 + 5 (0) 8 27 2 () 0 + (2) 7 ( + 2) + 2 () + ( +) ( ) ( +) + (4) 2 9 5 2 5 + (5) y 2 25 2 8y w 9y 2 4y 2 w (6) y 4 w 2 s s 5 w 2 s 2 (7) y 5w 8 2 25w 4 y 5w 2 y (8) 2 y 8 2 5w 25w 4 y 5w2 y
SIA(.) Sums and Differences of Rational Epressions Pages 87-9. On Pages 96-98 Problems.8,.9,.0,.a-g,. k,m,n. Perform the indicated operations and simplify. () 2 + 7 (2) + 0 () 2 + + (4) 2 + 4 + (5) + (6) + + + 2 2 (7) ( ) + (8) + ( ) + (9) ( +) ( ) + ( +) ( 2) (0) + 2 + 2 + + () 0 7 8 (2) +0 7 8 () 9 + 5 (4) 2 2 (5) ( +) 2 2 2 9 + (6) 2 2 9 2 (`7) + 4 4 (8) 4 7 4 + (9) 2 + + 2 2 (20) y y + 7 4 y (2) 2 + 2 + CCA(.) Displaying Functions Page 22 Problems, 2,. Problem 4 (The zeros are algebraically obvious. Use the calculator as an alternate approach, Prob 5-2. 4. Use your calculator to graphically find the zeros of f () = 0 + 7 (In this eample the zeros are essentially not accessible by algebraic means.)
CA(.2) Definitions (Functions) Pages 4-7. Problems 4-0, Problems -9. 20 and 2 (substitute problems) The notation f () or h() is inappropriate unless the relations are in fact functions. Replace these symbols by simply y. Then they represent relations between and y. Do these relations define y as a function of? If the answer is yes, state the domain and range of the function. 20. 2 0-2 -2 5 y 2 5 4 y 2 4 5 4 2 Problem 22-25. SIA(8.) Relations and Functions Pages 285-288. Set notation is not necessary. Instead charts and equations will be used to define functions and relations. For eample, if the tet uses {(, y) y = 2 + 4 } to define a function, it simply means the function y = 2 + 4. With this in mind: On Page 4 Problem 8. Find the domain of each of the following functions. () f () = 2 9 (2) g() = 2 25 () h() = 4 (4) g(t) = t +7t 2 + 5t 8 (5) f () = 4 ^2 2 (6) f () = + 7 (7) H() = ( + 4)( 8) (8) f () = 2 6 2 + 25 (9) f () = + 7
SIA(8.) Distance and Slope Formulas Pages 29-294. On pages 5-6 Problems 8.7, 8.8. A. Find the distance between the following pairs of points. Epress your answer algebraically and as a decimal. () (5,7) and (2,6) (2) (4,5) and (2, 6) () (-,5) and (-7, -) (4) (-9,-5) and (-, 2) (5) (-8,) and (4,-) (6) (8,2) and(-6, -4) B. Find the slope of the line through the following pairs of points. () (5,7) and (-2,6) (2) (4,5) and (2, 6) () (-,5) and (-7, -) (4) (-9,-5) and (-, 2) (5) (-8,) and (4,-) (6) (8,2) and(-6, -4) SIA(8.4) Linear Equation Forms Pages 295-02. On Pages 6-8 Problems 8.9, 8., Problem 8.0 (Determine the slope intercept form for the equation instead of the requested form), Problems 8.2, 8.. A. In each of the following, write the equation in slope intercept form of () The line through (5,9) with slope 4 (2) The line through (2,-7) with slope 6 () The line through (-4, 2) with slope - (4) The line through (-6,-8) with slope 4 (5) The line with -intercept - and slope 7 (6) The horizontal line through (5,-6) (7) The line through (4,5) with y-intercept -6 (8) The line through (5,7) and (-2,6) (9) The line through (-2,-8) with -intercept 5 (0) The line through (2,5) with (2,6) () The line with -intercept 4 and y-intercept - (2) The line through (5,-) and (,-) B. Find the slope and y-intercept of each of the following equations. () y = 8 (2) y = 5 4 () y = 2 ( 5 +7 ) (4) y = (5) 4 y + 9 = 0 (6) 2 + 5y = (7) 6y 2 = 9 (8) 8 + 5y = 2( y) + (9) = 4(y + 7)
SIA(8.5) Types of Functions Pages 02-07. On Pages 8-20 Problems 8.8, 8.9, 8.20. Find the linear function h() for which (a) h(5) = 6 and h( ) =2 (b) h(2) = 0 and h(0) =7 2. Find the quadratic function f () with f (0) = 2 and whose verte is at (-,5). Epress your answer in standard form, that is to say in the form f () = a 2 + b + c. Find the quadratic function f () with f () = 2 and whose verte is at (6,-5). Epress your answer in standard form (see prob 2) 4. Find the quadratic function h() with h(7) = 9 and whose verte is at (0,2). Epress your answer in standard form (see prob 2) 5. Find the quadratic function p() with p() =0 and whose verte is at (9,20). Epress your answer in standard form (see prob 2) 6. Graph the function p() = 2 2 +9 by first completing the square to find the verte. 7. Graph the function g() = 2 +4 58 by first completing the square to find the verte. 8. Epress the quadratic q() = 6 2 in verte form by completing the square. 9. Epress the quadratic q() = 5 2 8 + 7 in verte form by completing the square. 0. Epress the quadratic f () = 4 2 in verte form by completing the square.. Graph the following functions. (a) f () = + 5 (b) f () = 2 ( + )2 5 (c) h() = 2 4 (d) g() = 2 + + (e) f () = 4 (f) H() = 2 2. Graph the following functions (a) f () = 4 2 2 + > 2 (b) f () = 2 2 + 4 < (c) f () = 4 + 2 < 0 2 0 (d) g() = 4 >
SIA(8.7) Inverse Relations and Functions. Pages 09-. On Page 2-2 Problems 8.27, 8.28, 8.29 (skip part a). A Show that the following pairs of functions are inverses of each other. () f () = 7 8 and g() = 7 + 8 7 (2) p() = 2 + and q() = 2 () g() = 2 and h() = 2 + (4) g() = 2( + 7) and f () = 2 7 B Determine the inverse of each of the following functions. () f () = 5 + (2) g() = 8 5 () p() = (4) h() = 5 2 +, ( 0) (5) f () = (6 4) (6) h() = (6) 4 C Use your graphing calculator to determine which of the following functions have inverses that are functions. Justify your answer. (You do not have to find the inverse if it eists.) () f () = 2 2 + (2) h() = + 7 () g() = 25 2, ( 5 5) (4) p() = 7 8 (5) f () = 2 + 4 (6) h() = 7 2 8 + SIA(9.) Eponential Functions Pages 24-27. On Pages 50-5 Problems 9. and 9.2 A Graph the following without a calculator by making a chart of values and plotting points. () y = 4 2 (2) y = () y = 2 (4) y = 2 + (5) y = (6) y = 2 (7) y = 2 (8) y = 5 25
B Solve the following equations without a calculator by using eponential properties. Eplain your answer. () 2 = 64 (2) 5 = 25 () 9 = 27 (4) 5 = 8 (5) 2 =6 (6) 4 2 = 2 (7) 25 = (8) 4 + = 2 SIA(9.2) Logarithmic Functions. Pages 28-2. On Pages 5-54 Problems 9., 9.4, 9.5, 9.6 abc, 9.7, 9.8 (a,c,e,g) A Convert the following to logarithmic form. () 4 = 64 (2) 0 =.00 () 5 = (4) e 5 = B Convert the following to eponential form. () log 2 () =4 (2) ln(5) = () log 0 () = 54 (4) log 0 (.000) = 4 C. Evaluate the following without a caculator using the idea of a logarithm. Eplain your answer. () log 2 (2) (2) log 5 (25) () log 0 (.0) log 4 (2) C Find without a calculator using the idea of a logarithm. Eplain your answer. () log 2 () = 2 (2) log 0 () = () log 2 () = 7 (4) log 4 = 2 SIA(9.) Logarithmic Properties. Pages - 6, On Pages 52-5, Problems 9.9, 9.0. A Evaluate the following with a calculator using 6 digits of accuracy. () log 2 (.5) (2) log 5 (57.8) () log 6 (0.00458) (4) log 5 (.8) (5) log 4 (2) 2log 4 () (6) 2log 7 () + 5log 7 (2)
SIA(9.4) Epnential and Logarithmic Equations. Pages 7-42, (skip Solved Problem 9.20) On Page 55-56 Problems 9. Solve the following equations for using 6 digits of accuracy. () 6 = 7 (2) 8 = 0.000257 () 5 =7 (4) 2 5 = 82 (5) 4 = 78 (6) 5 = 8 2 (7) log 5 () = 4 (8) 2log 0 () = 7 (9) log ( + 4) + log (2) = (0) log 0 ( + 5) log 0 (4) = 2 () log 2 (4 ) + = log 2 (2) Special Section: Data Points and Eponential Functions. (see the supplement on the Math 22 web page. Go to the bottom of the main page under the heading Classroom Supplements and click on the link.) A In each of the following find the eact eponential function which passes through the given pair of data points. () ( 2,.4) and (, 4.) (2) (, 4) and (6, 7) () (, 9) and (5, 7) (4) (2, 8.) and (6,.9) (5) (.7, 9.2) and (5.,.8) B Determine (conclusively) whether there is an eponential function which passes through the three given points (with at least 6 digits of accuracy). If there is, determine its formula. () (2, ), (4, 4), (5, 9) (2) (, 27), (5, 6), (7, 2) () (2, 24), (4, 54), (5, 8) Special Section: The Basic Shapes of Power Functions: (see the supplement on the Math 22 web page. Go to the bottom of the main pageunder the heading Classroom Supplements and click on the link.) A. For each part, use a graphing calculator to graph on the same coordinate system y = k p for all of the values k and p listed. Clearly identify which equation belongs to each curve. () k = 2, p =, 2, (2) k=, p = /4, 2/4, /4 () k = 2, p = -, -2, - (4) p=, k =,, 6 (5) p = 2, k = /2, /.,4 (6) p = -2, k = 2, 5, 0 CCA(.) Predictions Based on Data, Pages 8-45 are worth reading and/or reviewing. Skip periodic functions on page 46. On pages 46-47 problems 2,, and Problem 4 (skip D, F) are good warm up eercises. Most of the problems in this section are regression problems done by use a graphing calculator to get a function whose curve best fits the data points in a given table. To get these problems see the regression link that appears on the main Math 22 web page under the heading, Suggested Problems for Math 22.
CCA(.4) Shifting and Scaling Pages 5-62. A graphing calculator should not be used in this section. Problem (Correction: the points listed in parts a, b, c are suppose to be the verte of the shifted graph of f () = 2 ). Problem 2, Problem 4-9, 9, 20. A In each of the following, the graph of the given function is shifted and or scaled as indicated. Give the formula which represents the graph. Graph the original function and the transformed function on the same coordinate system. You may use a graphing calculator if you wish. () f () = 2 2 shifted 2 units up and units to the left. (2) f () = 2 4 shifted units down and unit to the left. () g() = 5 /4 shifted 4 units up and 2 units to the right. (4) h() = shifted 2 units down and units to the right. (5) f () = stretched by a factor of 2 in the y direction. (6) f () = compressed by a factor of 2 in the y direction. (7) f () = stretched by a factor of 2 in the direction. (8) f () = compressed by a factor of 2 in the direction. (9) f () = stretched by a factor of in the y-direction (0) f () = () f () = stretched by a factor of in the -direction reflected about the -ais (2) f () = reflected about the y-ais
B The graph shown in Fig B below is the graph of a function f (). The figures B2 through B7 shown below are various transformations of the graph of f (). Match the function shown below with the figure that represents its graph. If there is no match state that as your answer. (The figures continue on the net page.) () 2 f () (2) f ( 4) + 20 () f ( + 4) + 20 (4) f 2 (5) 2 f () (6) f ( ) (7) f (2) (8) f () Fig B Fig B2 Fig B Fig B4
Fig B5 Fig B6 Fig B7 Fig B8
CCA(.5) and SIA(8.2) Algebra of Functions. CIA(.5) Pages 6-78, Problems 2, 4, 6,, 2 SIA 8.2 Pages 288-29, On SIA page 5 Problems 8.5, 8.6 A Determine and simplify the formula for f + g, f g, f g, f g, f o g, g o f () f () = 2 2, g() = 4 (2) f () = 2 + 7, g() = 9 () f () = 2 8, g() = 2 + 4 (4) f () = log ( 2 + 4), g() = 5 + 2 (5) f () = 2, g() = log ( + 2) (6) f () = 2 7, g() = 7 log 2 () CCA(.6) Graphical Approimations Pages 80-99. Problems 4-7, 9,, 4. On 9,, 4 use graphical techniques rather than algebraic techniques to approimate the answers to the questions that are asked. A. In each of the following a function f() is given and a point near a solution of the equation f() = 0. Approimate the solution to the equation near the given number to the nearest hundredth (2 digits to the right of the decimal point) using graphical techniques. Show a graph of f() near the solution point and offer convincing evidence that your solution is correct with to the nearest hundred. () f () = 8 2 + 2 + 25, = 2 (2) f () =0 2 2, = B Follow the same instructions as in Part A, but approimate with 4 digits of accuracy. As in Part A offer convincing evidence that your solution is correct with at least 4 digits of accuracy. () f () = 4 + 9, = 2 (2) f () =4 + log 0 (), =5 C The functions listed below are the same functions as in Part A. Find the remaining solutions to the equations f() = 0 using graphical techniques. Approimate to the nearest hundredth. Eplain why you know you have all of the solutions. Use the same techniques as in Parts A and B to show that your solutions are correct to the nearest hundredth. () f () = 8 2 + 2 + 25, = 2 (2) f () =0 2 2, = D Use graphical techniques to approimate the point of intersection of the graphs of the given pair of functions to the nearest hundredth. Use the same techniques as in the previous parts to show that your solutions are correct to the nearest hundredth. () f () = 2 + 2 + 5, g() = + 7 + 2 (2) f () = + 2, g() = +0
E. Aproimate to the nearest hundredth, the solutions to the following equations by using graphical techniques. Use the same techniques as in the previous parts to show that your solutions are correct to the nearest hundredth. () + 2 2 + 4 + 7 = 0 (2) = 2 2 6