PART I. Multiple choice. 1. Find the slope of the line shown here. 2. Find the slope of the line with equation $ÐB CÑœ(B &.

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1 Math College Algebra Final Exam Review Sheet Version X This review, while fairly comprehensive, should not be the only material used to study for the final exam. It should not be considered a preview of the final exam. It does not substitute for studying previous tests, quizzes, homework, class notes, text discussions, etc. There may be questions on the final exam unlike questions on this review, and vice versa. No question on this review will be exactly duplicated on the final exam. This review is longer than the final exam. You may find the following formulas helpful: Bœ È,,, %+- rise C C + Bœ + 7œ run œ B B " " PART I. Multiple choice. 1. Find the slope of the line shown here. % a. % b. % c. d. % 2. Find the slope of the line with equation ÐB CÑœ(B &. & % a. b. c. % d. 3. The following function relates the profit T (hundreds of dollars) to the number of hundreds of Itchy-Palm Hand-Held Computers B that are manufactured. T œ &ÐB )B "&Ñ Find the smallest break-even point for these computers (i.e. What number of computers makes the profit zero?). a. "& b.!! c. &ß!!! d. & 4. Analytically (by hand) solve the equation B œ &B ". & È" & È* È* a. B œ b. B œ &ß B œ " c. B œ d. B œ &

2 5. Which of the lines graphed here has negative slope? a. b. c. d. e. 6. Which of the following statements is true about the graph shown here? (There is only one correct answer.) a. The C-intercept is %. b. The graph is a parabola with vertex at the origin. c. The graph cannot be used as a function. d. The B-intercept is. e. The graph can be used as a function. 7. What is the abstract domain of the function 0ÐBÑ œ ( B '? a. all real numbers except ' b. all real numbers except c. all real numbers d. all real numbers except ( 8. Given that 0ÐBÑ œ B &, find 0Ð0Ð!ÑÑ. a. & b.! c.! d. "& 9. Given that 0ÐBÑ œ B &, find 0Ð2 + Ñ. a. 2 ( b. 2 ( c. 2 * d. 2 &

3 10. Find the range of the function graphed here. a. ÒßÑ b. Ð %ß'Ó c. Ð_ß_Ñ d. ÒßÓ 11. For the function 0 graphed here, determine 0ÐÑ. a. % b. c. % d. ) 12. If we assume that the function 0 graphed in Problem 11 is a polynomial function, what is the smallest possible degree the polynomial could have? a. & b. c. d. % 13. Which of the following numbers is not a root input of the function 0 graphed in Problem 11? a.! b. c. Þ& d. Þ& 14. What is the set of lower inputs of the function 0 graphed in Problem 11? a. Ð _ß!Ñ b. Ð _ß Þ&Ñ Ð!ß Þ&Ñ c. Ð!ß Þ&Ñ d. Ð Þ&ß!Ñ ÐÞ&ß _Ñ 15. The function 0 graphed in Problem 11 is decreasing over which interval of inputs? a. Ð Þ&ß!Ñ b. Ð _ß Ñ Ðß _Ñ c. Ð%ß %Ñ d. Ð ß Ñ 16. Which of the following points is a turning point of the function 0 graphed in Problem 11? a. Ð %ß &Ñ b. ÐÞ&ß!Ñ c. Ð ß %Ñ d. Ð!ß!Ñ 17. Evaluate JÐB "Ñ given that JÐ BÑ œ B BÞ a. B B " b. B B " c. B B " d. B B

4 18. Consider the following system of equations. Ú +, - œ ' Û Ü +, -œ) +, -œ" Is Ð+œ"ß,œ ß-œÑ a solution to this system? a. Yes b. No c. Cannot be determined 19. A parabola passes through the three points Ð "ß &Ñ, Ð"ß "Ñ and Ðß ÑÞ Set up a system of equations to find the coefficients of the symbol rule of the parabola. Ú +, *-œ& Ú +, -œ& a. Û +, -œ" b. Û +, -œ" Ü+, - œ Ü*+, - œ Ú +, -œ& Ú +, -œ& c. Û +, -œ" d. Û +, -œ" Ü*+, - œ Ü +, - œ 20. Consider the set WœÖBlB & and B &. Which number below is not an element of W? (There is only one correct answer.) "Î a. b.! c. " d. "' e. ' 21. An antique chair is expected to increase in value by! each year from 2000, when it was worth!!. If we be the value of the chair and > be the elapsed time in years from 2000, which function shown here can be used to predict the value of the chair in the future? œ!!> œ!!>! œ "!>!! œ!>!! 22. Find the C-intercept of the exponential function C œ Ð"Þ&Ñ B. a. b.! c. %Þ& d. "Þ& 23. The graph of the function 0ÐBÑ œ B %B is shown here. Use this graph to find the solution set of the inequality B %B!. (Answers are given in interval notation.) a. Ð!ß%Ñ b. Ð _ß!Ñ Ð%ß_Ñ c. Ð %ß_Ñ d. Ð _ß "Ñ Ð&ß _Ñ

5 24. What is the graphing window used in Problem 23? a. Ò _ß _Ó Ò _ß _Ó b. Ò(ß Ó Ò)ß 'Ó c. Ò ß (Ó Ò 'ß )Ó d. Ò Þ&ß (Þ&Ó Ò 'ß )Ó 25. What is minimum output of the function 0 graphed in Problem 23? a. none b. ' c. % d.! ( 26. Analytically solve the equation B ' œ. a. Bœ b. Bœ " c. Bœ d. Bœ 27. Which of the following numbers is an integer? "Î "Î " a. Î"! b. ( c. ") d. & 28. Which of the following numbers is not a real number? a. È b. È c. È% " ) % d. & 29. Evaluate 691 )". a. ( b. " c. % d. % 30. Evaluate 691 " "!!. a. b. & c. d. "! 31. Evaluate 691 % Ð "'Ñ. a. b. ' c. % d. 32. Find the domain of the function graphed here. a. Ð _ß Ñ b. Ò ß %Ó c. Ð _ß _Ñ d. Ð ß _Ñ 33. For the exponential function C œ Ð"Þ&ÑB, evaluate CÐ "Ñ. a. %Þ& b.!þ c. "Þ& d.

6 PART II. Show your work as appropriate. 34. Gail's Gourmet Bagels (GGB) is benefiting from the recent bagel craze by expanding its number of franchises. The following table contains data relating the number of GGB shops to the company's average weekly sales revenue (in thousands of dollars). Number of Shops Ð=Ñ "! "( "( %& %( & '! (& )& (current number) Average Weekly Revenue Ð<Ñ ""& ")! "*! &! '! %(! %' && '& (&& )'& 34a. Plot this table on the set of axes provided. 34b. Graph the straight line containing the points Ð"!ß ""&Ñ and Ð)&ß )'&Ñ. Does this line appear to fit this set of data points? Yes No 34c. Use the line drawn in part 34b to predict the average weekly sales revenue for GGB six months in the future, provided GGB is planning to open & more stores (above the current number) during this time period. 34d. Find the slope of the line drawn in part 34b. slope œ 34e. Interpret the slope of the line drawn in part 34b using a complete sentence. 34f. Analytically find the C-intercept of the line drawn in part 34b. (Do not estimate from the graph.) C-int. œ 34g. Let < be the output variable and let = be the input variable. Write the function symbol rule for the line drawn in part 34b. 34h. Express in function notation the average weekly sales revenue for GGB, provided "!! is the number of stores. Do not evaluate.

7 34i. Can the original table of data be used as a function? Yes No Explain. 35a. Does the line with the symbol rule B C œ ( contain the point Ðß Ñ? Yes No Explain. 35b. Is the symbol rule for the line shown in part 35a written in explicit form? Yes No Explain. 35c. Write the symbol rule for the line containing the point Ð %ß *Ñ that is parallel to the line shown in part 35a. 36. Solve the inequality ÐB Ñ Ÿ (ÐB "Ñ analytically. 37. Water usage in Gotham City has been increasing by %% per year since 1998, a trend that is expected to continue until In 1998, water usage was!! million gallons. Let's assume the function [ Ð>Ñ outputs the water usage in Gotham City (in millions of gallons) during year >, where > is the elapsed number of years since a. Find how much water was used in Gotham City in 1999; then express this information in function notation. 37b. Why is it appropriate to assume that [ is an exponential function? Briefly explain. 37c. Find the initial value - and growth factor (or base), for the function [. -œ,œ 37d. Write the symbol rule for [. [Ð>Ñ œ 37e. When the water usage in Gotham City reaches (!! million gallons, the city will need to switch from water wells to surface water. Write the equation that can be solved to predict the year when water usage will reach (!! million gallons. Do not solve the equation.

8 37f. Plot the application domain for the function [ on a number line. 38. A parabola has its vertex at the point Ðß %Ñ, its C-intercept is at ', and its two B-intercepts are at approximately " and &. Choose an appropriate graphing window for this parabola. ß ß &> 39. Consider the function 2Ð>Ñ œ ) ; is the value " in the range of this function? Yes No ExplainÞ 40. Consider the quadratic function Cœ1ÐBÑœB B '. 40a. Analytically find the B-intercept(s) of this function. B-int(s). œ 40b. Analytically find the coordinates of the vertex of this function. vertex œ 40c. Carefully sketch a graph of the function 1 on the set of axes given below; a table has been provided for your convenience to compute coordinates of points on the parabola. B C 40d. Use the graph in part 40c to find the set of upper inputs of 1. Write this set using interval notation. 40e. Use the graph in part 40c to solve the inequality B B '!. Write the solution set using interval notation.

9 41. Suppose you are trying to find an approximate solution to the equation 41a. Write an associated function for this equation. 0ÐBÑ œ % B 'B B œ &!B 41b. An associated function 0 for the given equation has been tabulated below with two different table domains. Which one of the tables is most useful to determine a solution to the given equation? Circle your answer. i) B %Þ%" %Þ%" %Þ%"% %Þ%"& %Þ%"' ii) B "Þ!!!!Þ&!!!!Þ&!! "Þ!!! 0ÐBÑ!Þ!*!Þ!%*!Þ!!)!Þ!!Þ!( 0ÐBÑ &' %)Þ') &!Þ") ) ii) From the table circled above, estimate a solution correct to two decimal places. solution œ 42. Find the C-intercept of the function described in Problem 3. C-int. œ Interpret this value in the context of the problem. 43. For the polynomial function 1ÐDÑ œ D D D : 43a. What is the degree of 1? degree œ 43b. What is the maximum possible number of B-intercepts of 1? 43c. What is the maximum possible number of turning points of 1? 43d. What is the leading coefficient of 1? 43e. What is the C-intercept of 1? 43f. Is the number " a root input of 1? Yes No 43g. Is the number a lower input of 1? Yes No 43h. Is the number! an upper input of 1? Yes No 43i. Write the abstract domain of 1Þ domain œ 44. Explain why each of the following functions is not a polynomial function. 44a. :ÐBÑ œ È B " 44b. 2Ð7Ñ œ È " The quantity C is directly proportional to the quantity B, and C œ "& when B œ. Find the constant of proportionality 5 and write the symbol rule for a function that can be used to determine the value C based on the number B. Cœ 46. The quantity C is inversely proportional to the quantity B, and C œ "& when B œ. Find the constant of proportionality 5 and write the symbol rule for a function that can be used to determine the value C based on the number B. Cœ

10 47. Which of the following equations and inequalities appear to be more easily solved by analytical methods? Which appear to be good candidates for solution by numerical or graphical methods (i.e. through the use of an associated function)? 47a. ÐB (Ñ œ BÐB %Ñ 47b. ÐB (Ñ BÐB %Ñ 47c. ÐB (Ñ ÐB %Ñ 47d. B ( œ B % 47e. B ( B % B ( B % 47f. Ÿ % 47g. B (B B %B " œ! % 47h. B (B B œ! 47i. l lðb (Ñ œ l lðb %Ñ 47j. l B ( l œ l B % l 48. Use the Quadratic Formula to show that the function 1ÐBÑœÐB "Ñ (B does not have any B-intercepts. % 49. Estimate one solution to the following equation by tabulating an associated function: B (B B %B " œ!. Write your solution correct to three decimal places. solution œ 50. Estimate the solution to the following inequality by graphing an associated function: ÐB (Ñ BÐB %Ñ. solution œ 51. Express the solution to each of the following equations in terms of logarithms. Then use a calculator or computer to estimate the solution rounded to two decimal places. B 51a. "! œ *! B œ B 51b. œ *! B œ B 51c. / œ *! B œ 52. Let Ú B "ß if B! 0ÐBÑ œ Û ß if! Ÿ B Ÿ % Ü % Bß if B % 52a. Evaluate: 0ÐÑ œ 52b. Evaluate: 0Ð&Ñ œ 52c. Complete the statement. The domain of 0 is the interval Ð ß Ñ.