3 ÍNDICES DE DESARROLLO LOCAL (IDL)
3.5 Dimensión Educación
A special linear function that has the form y = 0x + b, or y = b, where b is a real number, is called a constant function. The graph of the constant function y = 3 is shown in Figure 1.36(a). The temperature inside a sealed case containing an Egyptian mummy in a museum is a constant function of time because the temperature inside the case never changes. Notice that even though the range of a constant function consists of a single value, the input of a constant function is any real number or any real number that makes sense in the context of an applied problem. Another special linear function is the identity function
y = 1x + 0, or y = x
which is a linear function of the form y = mx + b with slope m = 1 and y-intercept
b = 0. For the general identity function f (x) = x, the domain and range are each the
set of all real numbers. A graph of the identity function f is shown in Figure 1.36(b).
1. Which of the following functions are linear? a. y = 3x2 + 2 b. 3x + 2y = 12 c. y = 1
x + 2
6. Find the slope of the line in the graph below.
Skills CHECK
1.3
2. Is the graph in the figure below a function?
4. Find the slope of the line through (8, -10) and (8, 4). 5. Find the slope of the line in the graph that follows. 3. Find the slope of the line through (4, 6) and (28, -6).
5 4 3 2 1 0 1 2 3 4 5 5 4 3 2 1 1 2 3 4 5 x y 5 4 3 2 1 0 1 2 3 4 5 6 4 2 2 4 6 8 10 x y 5 4 3 2 1 0 1 2 3 4 5 x y 5 4 3 2 1 1 2 3 4 Constant Function (a) Figure 1.36 4 2 2 4 2 4 0 2 4 y x y 3 3 2 1 0 1 2 3 3 2 1 1 2 3 x f(x) f(x) x Identity Function (b)
28. If a linear function has the points (2, 1) and (6, 3) on its graph, what is the rate of change of the function?
17. 5y = 2 18. x = 6
In Exercises 7–10, (a) find the x- and y-intercepts of the graph of the given equation, if they exist, and (b) graph the equation.
7. 5x - 3y = 15 8. x + 5y = 17
For Exercises 15–18, (a) give the slope of the line (if it exists) and the y-intercept (if it exists) and (b) graph the line.
15. y = 4x + 8 16. 3x + 2y = 7 9. 3y = 9 - 6x 10. y = 9x 6 4 2 0 4 6 5 10 5 10 x y
11. If a line is horizontal, then its slope is ____. If a line is vertical, then its slope is ____.
12. Describe the line whose slope was determined in Exercise 4.
For Exercises 13–14, determine whether the slope of the graph of the line is positive, negative, 0, or undefined.
13. a. 2 0 2 4 6 8 6 10 x 4 2 2 4 6 8 y 2 0 2 6 4 6 10 5 10 x y 2 0 2 4 4 6 5 10 10 x y b. 14. a. b.
For each of the functions in Exercises 19 –21, do the following:
a. Find the slope and y-intercept (if possible) of the
graph of the function.
b. Determine if the graph is rising or falling.
c. Graph each function on a window with the given
x-range and a y-range that shows a complete graph.
19. y = 4x + 5; [-5, 5]
20. y = 0.001x - 0.03; [-100, 100] 21. y = 50,000 - 100x; [0, 500]
22. Rank the functions in Exercises 19–21 in order of increasing steepness.
For each of the functions in Exercises 23–26, find the rate of change.
23. y = 4x - 3 24. y = 1
3 x + 2
25. y = 300 - 15x 26. y = 300x - 15
27. If a linear function has the points (-1, 3) and (4, -7) on its graph, what is the rate of change of the function?
29. a. Does graph (i) or graph (ii) represent the identity function?
b. Does graph (i) or graph (ii) represent a constant function? y 3 1 0 1 2 3 2 3 2 1 3 1 2 3 x y (i) f(x) x 1 0 1 2 3 2 3 2 1 3 1 2 3 x f(x) (ii)
30. What is the slope of the identity function?
31. a. What is the slope of the constant function y = k?
32. What is the rate of change of the identity function?
b. What is the rate of change of a constant func- tion?
33. College Enrollment The total fall enrollment in 4-year public institutions for the years 1990 through 2008 is given by y = 0.014x + 2.290, where x is the number of years after 1990 and y is millions of students. Is this a linear function? Why or why not? What is the independent variable?
(Source: U.S. Department of Education, National Center for Education Statistics)
a. Find the x-intercept of the graph of this function.
EXERCISES
1.3
34. Women in the Workforce Using data and projections from 1950 through 2050 gives the number y (in thousands) of women in the workforce as the function y = -0.005x2+ 1.278x + 13.332, where x is the number of years from 1950. Is this a linear function? Why or why not?
(Source: U.S. Census Bureau, U.S. Dept. of Commerce)
35. Marriage Rate The marriage rate per 1000 pop- ulation for the years 1987–2009 is given by
M(x) = -0.146x + 11.074, where x is the number
of years after 1980.
a. Why is this a linear function, with y = M(x)? b. What is the slope? What does this slope tell you
about the number of unmarried women who get married?
(Source: National Vital Statistics Report)
36. Prescription Drug Sales Retail prescription drug sales for the years 1995–2005, in billions of dollars, can be modeled by the function
y = 14.232x + 8.073
where x is the number of years after 1990.
a. Why is this a linear function?
b. What is the slope of the graph of the function? c. What is the rate at which the sales grew during
this period?
(Source: U.S. Census Bureau)
37. Marijuana Use The percent p of high school seniors who ever used marijuana can be related to
x, the number of years after 2000, by the equation
25p + 21x = 1215.
b. Find and interpret the p-intercept of the graph of this function.
c. Graph the function, using the intercepts. What val- ues of x on the graph represent years 2000 and after? 38. Depreciation An $828,000 building is depreciated
for tax purposes by its owner, using the straight-line depreciation method. The value of the building after
x months of use is given by y = 828,000 - 2300x
dollars.
a. Find and interpret the y-intercept of the graph of this function.
b. Find and interpret the x-intercept of the graph of this function.
c. Use the intercepts to graph the function for non- negative x- and y-values.
39. Asparagus Cultivation In Canada, the cultivation area (in hectares) of asparagus between 1999 and 2008 is shown in the table.
Year 1999 2000 2001 2002 2003 Cultivation Area (hectares) 1200 1200 1200 1200 1200 (Source: Sustainablog.org) Year 2004 2005 2006 2007 2008 Cultivation Area (hectares) 1200 1200 1200 1200 1200
a. Sketch the data as a scatter plot, with y equal to the cultivation area and x equal to the year. b. Could the data in the table be modeled by a constant
function or the identity function?
c. Write the equation of a function that fits the data points.
d. Sketch a graph of the function you found in part (c) on the same axes as the scatter plot.
40. Life Insurance The monthly rates for a $100,000 life insurance policy for males aged 27–32 are shown in the table.
Age (years), x 27 28 29
Premium (dollars per month), y 11.81 11.81 11.81
Age (years), x 30 31 32
Premium (dollars per month), y 11.81 11.81 11.81
44. Diabetes The figure below shows the projected per- cent of U.S. adults with diabetes for the years 2010 through 2050. What is the average rate of growth over this period of time?
a. Could the data in this table be modeled by a constant function or the identity function?
b. Write an equation whose graph contains the data points in the table.
c. What is the slope of the graph of the function found in part (b)?
d. What is the rate of change of the data in the table? 41. Cigarette Use For the years 1965–2009, the percent
p of adults who have tried cigarettes can be modeled
by p = 43.3 - 0.504t, where t is the number of years after 1960.
a. Is the rate of change of the percent positive or negative?
b. How fast was the percent of adults who tried ciga- rettes during this period changing? Use the units in the problem in your answer.
(Source: monitoringthefuture.org)
42. Crickets The number of times per minute n that a cricket chirps can be modeled as a function of the Fahrenheit temperature T. The data can be approxi- mated by the function
n = 12T
7 - 52
7
a. Is the rate of change of the number of chirps posi- tive or negative?
b. What does this tell us about the relationship between temperature and the number of chirps? 43. Twitter Between 2007 and 2010, “tweeting” became
increasingly popular. The number of tweets (in thousands) reported by the microblogging com- pany Twitter can be modeled by T(x) = 16,665x - 116,650, where x is the number of years after 2000. a. What is the slope of the graph of this function? b. Interpret the slope as a rate of change.
c. According to Computerworld, the number of tweets grew by 1400% from 2009 to 2010. Do your results from (b) confirm this?
(Source: Centers for Disease Control and Prevention)
0% 10% 5% 15% 25% 20% 30% 35% 2010 2015 2020 2025 2030 (2010, 15.7) 2035 2040 (2050, 34.3) 2050 2045
Percent with diabetes (estimated)
45. Earnings and Minorities According to the U.S. Equal Employment Opportunity Commission, the relation- ship between the median annual salaries of minori- ties and whites can be modeled by the function
M = 0.959W - 1.226, where M and W represent
the median annual salary (in thousands of dollars) for minorities and whites, respectively.
a. Is this function a linear function?
b. What is the slope of the graph of this function? c. Interpret the slope as a rate of change.
(Source: Statistical Abstract of the United States)
46. Marijuana Use The percent p of high school seniors using marijuana daily can be modeled by 30p - 19x = 30, where x is the number of years after 1990.
a. Use this model to determine the slope of the graph of this function if x is the independent variable. b. What is the rate of change of the percent of high
school seniors using marijuana per year?
(Source: Index of Leading Cultural Indicators)
47. Black Population Using data and projections from 1990 through 2050, the percent of the U.S. population that is black can be modeled by B(x) = 0.057x +12.3, where x is the number of years after 1990.
a. What is the slope of the graph of this function? b. Interpret the slope as a rate of change.
(Source: U.S. Census Bureau)
48. Seawater Pressure In seawater, the pressure p is related to the depth d according to the model 33p - 18d = 496, where d is the depth in feet and p is in pounds per square inch.
a. What is the slope of the graph of this function? property after x years and the line representing the value as a function of years passes through the points (10, 1,310,000) and (20, 700,000).
a. What is the slope of the line through these points? b. Interpret the slope as a rate of change.
49. Advertising Impact An advertising agency has found that when it promotes a new product in a city, the weekly rate of change R of the number of people who are aware of it x weeks after it is introduced is given by R = 3500 - 70x. Find the x- and R-intercepts and then graph the function on a viewing window that is meaningful in the application.
50. Internet Users The percent of the U.S. population with Internet access can be modeled by
y = 5.0x - 6.5
where x is the number of years after 1995.
a. Find the slope and the y-intercept of the graph of this equation.
b. What interpretation could be given to the slope?
(Source: Jupiter Media Metrix)
51. Call Centers Call centers are booming in the Philippines as multinational companies increasingly outsource these operations. The annual revenue for call centers in the Philippines from 2006 to 2010, in billions of dollars, can be modeled by R(x) = 0.975x -3.45, where x is the number of years after 2000.
a. According to the model, what was the rate of change of revenue for call centers in the Philippines?
b. According to the model, what was the revenue for call centers in the Philippines in 2010?
c. Would this model be valid to estimate the revenue in 2000? Why or why not?
(Source: Business Processing Association of the Philippines)
53. Depreciation Suppose the cost of a business property is $1,920,000 and a company depreciates it with the straight-line method. Suppose V is the value of the
b. What is the annual rate of change of the value of the property?
54. Men in the Workforce The number of men in the workforce (in millions) for the years from 1890 to 2008 can be approximated by the linear model deter- mined by connecting the points (1890, 18.1) and (2008, 67.8).
a. Find the annual rate of change of the model whose graph is the line connecting these points.
b. What does this tell us about men in the workforce? 55. Profit A company charting its profits notices that
the relationship between the number of units sold
x and the profit P is linear. If 300 units sold results
in $4650 profit and 375 units sold results in $9000 profit, find the marginal profit, which is the rate of change of the profit.
56. Cost A company buys and retails baseball caps, and the total cost function is linear. The total cost for 200 caps is $2690, and the total cost for 500 caps is $3530. What is the marginal cost, which is the rate of change of the function?
57. Marginal Cost Suppose the monthly total cost for the manufacture of golf balls is C(x) = 3450 + 0.56x, where x is the number of balls produced each month. a. What is the slope of the graph of the total cost
function?
b. What is the marginal cost (rate of change of the cost function) for the product?
c. What is the cost of each additional ball that is pro- duced in a month?
58. Marginal Cost Suppose the monthly total cost for the manufacture of 19-inch television sets is
C(x) = 2546 + 98x, where x is the number of TVs
produced each month.
a. What is the slope of the graph of the total cost function?
b. What is the marginal cost for the product? c. Interpret the marginal cost for this product. 59. Marginal Revenue Suppose the monthly total rev-
enue for the sale of golf balls is R(x) = 1.60x, where
x is the number of balls sold each month.
a. What is the slope of the graph of the total revenue function?
52. Wireless Service Spending The total amount spent in the United States for wireless communication ser- vices S (in billions of dollars) can be modeled by the function
S = 6.205 + 11.23t
where t is the number of years after 1995.
a. Find the slope and the y-intercept of the graph of this equation.
b. What interpretation could be given to the
y-intercept?
c. What interpretation could be given to the slope?
b. What is the marginal revenue for the product? c. Interpret the marginal revenue for this product.
61. Profit The profit for a product is given by P(x) =
19x - 5060, where x is the number of units produced and sold. Find the marginal profit for the product. 60. Marginal Revenue Suppose the monthly total revenue
from the sale of 19-inch television sets is R(x) = 198x, where x is the number of TVs sold each month. a. What is the slope of the graph of the total revenue
function?
b. What is the marginal revenue for the product? c. Interpret the marginal revenue for this product.
62. Profit The profit for a product is given by the func- tion P(x) = 939x - 12,207, where x is the number of units produced and sold. Find the marginal profit for the product.
1.4
Equations of Lines
KEY OBJECTIVES
■ Write equations of lines using the slope-intercept form and the point-slope form
■ Write equations of horizon- tal and vertical lines ■ Write the equations of lines
parallel or perpendicular to given lines
■ Find the average rate of change over an interval for nonlinear functions ■ Find the slope of the
secant line between two points on a graph ■ Find the difference
quotient from 1x, f (x)2 to 1x + h, f (x + h)2 ■ Find average rates of
change for approximately linear data