Answer Key for Exam A

Transcription

1 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 y 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

2 7. Crafty Bill s Cool Car Sales opened as a used car sales lot in 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 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() = 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

3 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 = by plotting points. Answer: 11. Graph the functiony = 3 with a graphing utility. 6 Answer: 3

4 12. The polynomial 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 returns million items. 13. The polynomial function I(t) = 0.1t t 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

5 14. Find the slope of the line through the pair of points (2, 6) and (1, 3) (d) 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 = (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

6 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 = y = y = Hence the slope m = 6 and they-intercept isb = Graph the equationy = Answer: y 6

7 19. The cost of tuition at a community college is given by C() = , 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 Let = 0 represent Find the equation giving yearly sales S(). S() = S() = S() = (d) S() = In one U.S. town the annual consumption, b, in beef (in pounds per person) can be estimated by b = t, where t is the number of years since 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 = 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 = ( 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 = y = y =

8 24. Find the average rate of change for the functiony = 2 +5 between = 2 and = (d) 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 = 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 Number Answer: In particular, the rounded linear regression is: y =

9 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) 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 (d) 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 Number (d) 16,795,800 products sold 69,540 products sold 72,530 products sold 79,240 products sold 9

10 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 = y = 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? (d) A manufacturer has total revenue given by the functionr = 170 and has total cost given byc = , whereis the number of units produced and sold. Find the break-even number of units for the manufacturer. (d) units 200 units 140 units units 10