Answer Key for Eam A For questions 1-2, determine whether the given relationship defines a function. Eplain your answer. 1. The temperatureton a backyard thermometer at 5 pm on a given day. Answer: This definestas a function of, since there can be only one temperature at 5 pm. 2. The temperatureton a backyard thermometer on a given day. Answer: In this case t is not a function of, since there are multiple temperature values that occur during a single day. 3. 1 9 1 7 y 2 1 3 8 Does the table definey as a function of? What aboutas a function ofy? Answer: y is not a function of, since = 1 gives two values (y = 2 andy = 3) ofy. However,is a function of y. 4. Decide whether or not the graph is or is not that of a function. y Yes No 5. Decide whether or not the set of unordered pairs defines a function: {( 3, 7),( 2, 2),(2,2),(7,2)} Yes No 6. Decide whether or not the equation = y 2 2 defines y as a function of. Yes No 1
7. Crafty Bill s Cool Car Sales opened as a used car sales lot in 1991. The graph shows the number of cars sold as a function of time. What is the domain of this function if we consider the indicated points? y 900 800 700 600 500 400 300 200 100 1991 1992 1993 1994 1995 Answer: The domain is the set of input values. In this case the domain is the closed interval[1991,1995] 8. Employees of a publishing company received an increase of salary of 7% plus a bonus of $900. Let S() = 1.07+900 represent the new salary in terms of the previous salary. Find an interprets(13000). (d) $13,900: If an employee s old salary was $13,000, then his/her new salary was $13,900 after the increase and bonus. $23,000: If an employee s old salary was $23,000, then his/her new salary was $13,000 after the increase and bonus. $11,308: If an employee s old salary was $11,308, then his/her new salary was $13,000 after the increase and bonus. $14,810: If an employee s old salary was $13,000, then his/her new salary was $14,810 after the increase and bonus. 2
9. The following graph shows the stock price of a new internet company over the first 18 months after the initial public offering of its stock. Stock Price (in dollars) Month Approimately in which month(s) did the stock price reach $60? The 2nd and 10th months The 10th and 18th months The price never reached $60 (d) The 18th month 10. Graph the functiony = 2 +2+1 by plotting points. Answer: 11. Graph the functiony = 3 with a graphing utility. 6 Answer: 3
12. The polynomial0.0031 4 +0.0051 3 +0.0041 2 +0.15+1.22 gives the predicted sales volume of a company, in millions of items, where is the number of years from now. Determine the predicted sales 20 years from now. Round your answer to the nearest hundredth million. Answer: When = 20, the polynomial returns542.66 million items. 13. The polynomial function I(t) = 0.1t 2 +1.4t represents the yearly income (or loss) from a real estate investment, wheretis the time in years. After what year does income begin to decline? Answer: Using the graphing function (and the Trace Etremey value tool): we see that t = 7 is the turning point of the income function, after which the income decreases. 4
14. Find the slope of the line through the pair of points (2, 6) and (1, 3). 1 3 3 1 3 (d) 3 15. Decide whether the slope is positive, negative, zero, or undefined. y (d) Zero Negative Undefined Positive 16. Find the- and y-intercepts of the graph of 3 18y = 18, if they eist. Then graph the equation. Answer: The-intercept occurs when y = 0: 3 18y = 18 3 18(0) = 18 3 = 18 = 6 so the -intercept is the point( 6,0). For the y-intercept, we set = 0: 3 18y = 18 3(0) 18y = 18 18y = 18 y = 1 so they-intercept in the point(0, 1). 5
y 17. Find the slope of the line 6 8y = 16 (if it eists) and the y-intercept (if it eists). Answer: We need to put the equation6 8y = 16 in slope-intercept (y = m+b) form. 6 8y = 16 8y = 6 16 y = 6 16 8 y = 6 8 16 8 y = 6 8 +2 Hence the slope m = 6 and they-intercept isb = 2. 8 18. Graph the equationy = 3 2 1. Answer: y 6
19. The cost of tuition at a community college is given by C() = 462 + 50, where is the number of credit hours. Find and interpret thec-intercept of the graph of this function. (d) 50: The tuition increases by $50 for each additional credit hour 462: The tuition increases by $462 for each additional credit hour 50: There is a tuition fee of $50 in addition to the charge per credit hour 462: There is a tuition fee of $462 in addition to the charge per credit hour 20. Assume that the sales of a certain appliance dealer are approimated by a linear function. Suppose that sales were $12,000 in 1982 and $54,000 in 1987. Let = 0 represent 1982. Find the equation giving yearly sales S(). S() = 42000+54000 S() = 8400+54000 S() = 8400+12000 (d) S() = 42000+12000 21. In one U.S. town the annual consumption, b, in beef (in pounds per person) can be estimated by b = 37 0.6t, where t is the number of years since 1975. What is the slope of the graph of this function? Write a sentence interpreting its value. Answer: The slope of the graph of this function is m = 0.6 which represents the fact that the number of pounds of beef each person consumes decreases by 0.6 each year. 22. Does every line have an -intercept? If not, give an eample of an equation whose graph does not have an -intercept. Answer: Any line that is parallel to the-ais does not have an -intercept. For eampley = 1. 23. Write the slope-intercept form of the equation for the line passing through( 7,6) and(1, 3). Answer: Call( 1,y 1 ) = ( 7,6) and( 2,y 2 ) = (1, 3). The point-slope formula is wheremis the slope of the line. So that the point-slope formula for this line is y y 1 = m( 1 ) m = y 2 y 1 2 1 = 3 6 1 ( 7) = 9 8 y 6 = 9 8 ( ( 7)) = 9 8 (+7) The slope-intercept form is obtained by solving fory: y 6 = 9 8 (+7) y 6 = 9(+7) 8 y 6 = 9 8 7 8 y = 9 8 7 8 +6 y = 9 8 + 41 8 7
24. Find the average rate of change for the functiony = 2 +5 between = 2 and = 9. 16 14 112 9 (d) 18 25. An electrician charges a fee of $40 plus $25 per hour. Let y be the cost in dollars of using the electrician for hours. Find the slope-intercept form of the equation. Answer: The rate (of change) is $25 per hour, while the fied cost (initial charge when = 0, i.e., the y- intercept) is $40: y = 25+40 26. The paired data below consists of the costs of advertising (in thousands of dollars) and the number of products sold (in thousands). Use linear regression to find a rounded linear function (to nearest hundreth thousand) that predicts the number of products sold as a function of the cost of advertising. Cost 9 2 3 4 2 5 9 10 Number 85 52 55 68 67 86 83 73 Answer: In particular, the rounded linear regression is: y = 2.79+55.79 8
27. The following graph shows data for a recent train ride from New York to Toronto. At What rate did the train travel? y Distance from New York (miles) 300 200 100 1 2 3 4 Time of Day (PM) 60 miles per hour 65 miles per hour 50 miles per hour (d) 120 miles per hour 28. Solve the equation 1 6 (+18) 1 ( 7) = +5 7 and then check your answer using a graphing utility. 133 55 119 55 203 55 (d) 294 55 29. The paired data below consists of advertising (in thousands of dollars) and the number of products sold (in thousands). Use linear regression to find a linear function, rounded to the nearest thousandth, that predicts the number of products sold as a function of the cost of advertising and predict the number of products sold (in thousands) if the cost of advertising is $6000. Cost 9 2 3 4 2 5 9 10 Number 85 52 55 68 67 86 83 73 (d) 16,795,800 products sold 69,540 products sold 72,530 products sold 79,240 products sold 9
30. Solve the system of equations using a graphing aid. { 3 2y = 1 3+4y = 29 Answer: The solution is the point in which the graphs of the two functions intersect : (3,5). 31. Solve the system of equations by hand using any method. { 9+6y = 33 2+4y = 2 = 3,y = 2 = 2,y = 2 = 3,y = 1 (d) No solution 32. Nadine sold two kinds of tickets to her class play. Student tickets cost $4.00 each, and adult tickets cost $6.50 each. If Nadine sold a total of 35 tickets for $182.50, how many student tickets did she sell? 18 22 17 (d) 20 33. A manufacturer has total revenue given by the functionr = 170 and has total cost given byc = 28900+30, whereis the number of units produced and sold. Find the break-even number of units for the manufacturer. (d) 144.5 units 200 units 140 units 206.4 units 10