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Unformatted text preview: g Section 32 GraphsandLines 25 3. (A) (B) Horizontal line: y = 2; vertical line: = —8
4. (A) 0 (B) 4 (C) Not deﬁned (D) 1 6.(A)2x3y=18
(B) y = 3x — 9
7. (A) m = 40
(B) C = 40x + 250
8. (A) p = 0.008): — 12.84 (B) p = 0.04x + 99.8
(C) (2,347, 5.93) Identify the graph(s) of lines with a negative slope. N'." . Identify the graph(s) of lines, with a positive slope. 9‘ Identify the graph(s) of any lines with slope zero. ,3 Identify the graph(s) of any lines with undefined slope.
What can you say about their slopes? In Problems 5—8, sketch a graph of each equation in a rectan
gular coordinate system. 5. y=2x—3 6. y=325+1
7. 2x+3y=12 s. 8x—37=24 In Problems 9—12, ﬁnd the slope and y intercept of the graph of
each equation. 9. y=3x+1 1o. y=35‘——2
11. y=—%x—6 12. y=0.7x+5 In Problems 13—16, write an equation of the line with the indi
cated slope and y intercept. 13. 81996 = 2 14. Slope =§
ylntercept = 3 y intercept = 5 15. Slope =§ 16. Slope = —5
yintercept = *4 y intercept = 9 26 C H A P T E R 1 Linear Equations and Graphs In Problems 1720, use the graph of each line to find the x
intercept, y intercept, and slope. Write the slopeintercept form
of the equation of the line. 17. y IIIIII
IIIIll.“ 3 Sketch a graph of each equation or pair of equations in
Problems 2126 in a rectangular coordinate system. 21.y=—§x2 22. y=—§‘x+1
23. 3x—2y=10
25. x=3;y=2 26. In Problems 27—30, ﬁnd the slope of the graph of each equation. 27. 4x+y=3
29. 3x+5y=15 28. 5x—y=—2
30. 21—3y=18 31. Given Ax + By = 12, graph each of the following three cases in the same coordinate system.
(A) A =2andB = 0 (B) A=0andB=3
(C) A=3andB=4 i
i
E
l
i
g
l
l
g
l 32. Given Ax + By = 24, graph each of the following three
cases in the same coordinate system. (A) A=6andB=0
(B) A=0andB=8
(C) A=2andB=3 33. Graph y = 251: + 200, x 2 0.
34. Graphy = 40x +160,x Z 0. 35. (A) Graph y = 1.2x — 4.2 in a rectangular coordinate
system. (B) Find the x and y intercepts algebraically to one
decimal place. (C) Graph y = 1.2x ~ 4.2 in a graphing utility. ate (D) Find the x and y intercepts to one decimal place
using TRACE and the zero command on your graphing
utility. 36. (A) Graph y = ~0.8x + 5.2 in a rectangular coordinate
system. (B) Find the x and y intercepts algebraically to one
decimal place. (C) Graph y = —0.8x + 5.2 in a graphing utility. (D) Find the x and y intercepts to one decimal place
using TRACE and the zero command on your graphing
utility. (E) Using the results of parts (A) and (B) or (C)
and (D), find the solution set for the linear
inequality @@ —0.8x + 5.2 < 0 In Problems 37—40, write the equations of the vertical and
horizontal lines through each point. 37. (4, —3) ’ 38. (5,6)
39. (—1.5,—3:5) 40. (26,38) In Problems 4146, write the equation of the line through each
indicated point with the indicated slope. Write the ﬁnal answer
in the form y = mx + b. 41. m = 4; (2, ~3)
43. m = §;(—4,—5)
45. m = O; (—1.5, 4.6) 42. m = —6; (—4, 1)
44. m = 3—; (—6, 2)
46. m = o; (3.1, —2.7) In Problems 47—54,
(A) Find the slope of the line that passes through the given points.
(E) Find the standard form of the equation of the line.
47. (2, 5) and (5, 7) 48. (1, 2) and (3, 5) 49. (—2, —1) and (2, —6) 50.
51. (5, 3) and (5, —3)
53. (—2, 5) and (3, 5) (2, 3) and (—3, 7) 52. (1,4) and (0,4) 54. (2,0) and (2, —3) 55. Discuss the relationship among the graphs of the lines
with equation y = mx + 2, where m is any real number. 56. Discuss the relationship among the graphs of the lines
with equation y = ——O.5x + b, where b is any real
number. A'gipliéations 57. 58. 59. 61. 62. 63. Simple interest. If $P (the principal) is invested at an interest rate of r, then the amount A that is due after t p years is given by
A = Prt + P
If $100 is invested at 6% (r = 0.06). then
A = 6t + 100,t _>_ 0 ( A) What will $100 amount to after 5 years? After 20 years? (B) Sketch a graph of A = 6r + 100 for 0 S t S 20.
(C) Find the slope of the graph and interpret verbally. Simple interest. Use the simple interest formula from Prob _2
lem 57. If $1,000 is invested at 7.5% (r = 0.075), then ‘ A=75t+1,000.t_>_0. (A) What will $1,000 amount to after 5 years? After : > 20 years?
(B) Sketch a graph of A = 75t + 1,000 for0 S t S 20.
(C) Find the slope of the graph and interpret verbally. Cost analysis A doughnut shop has a fixed cost of $124 _
per day and a variable cost of $0.12 per doughnut. Find the 1
total daily cost of producing x doughnuts. How many .
doughnuts can be produced for a total daily cost of $250? Cost analysis. A small company manufactures picnic tables "
The weekly fixed cost is $1,200 and the variable cost is i
$45 per table. Find the total daily cost of producing x pic ;
nic tables How many picnic tables can be produced for a total weekly cost of $4,800? Cost analysis A plant can manufacture 80 golf clubs per
day for a total daily cost of $7,647 and 100 golf clubs per ; day for a total daily cost of $9,147.
(A) Assuming that daily cost and production are linearly related, ﬁnd the total daily cost of producing x golf clubs , (B) Graph the total daily cost for 0 S x S 200. (C ) Interpret the slope and y intercept of this cost equation. Cost analysis A plant can manufacture 50 tennis rackets
[361' day for a total daily cost of $3,855 and 60 tennis rack ets per day for a total daily cost of $4,245. (A) Assuming that daily cost and production are linearly
related, find the total daily cost of producing x tennis 3 rackets.
(B) Graph the total daily cost forD S x S 100. (C) Interpret the slope and y intercept of this cost equation. 2
Business~Markup policy. A drug store sells a drug cost ing $85 for $112 and a drug costing $175 for $238. (A) If the markup policy of the drug store is assumed to
be linear, write an equation that expresses retail price R in terms of cost C (wholesale price). (B) What does a store pay (tothe nearest dollar) for a § drug that retails for $185? Business—Markup policy. A clothing store sells a shirt costing $20 for $33 and a jacket costing $60 for $93. (A) If the markup policy of the store is assumed to be lin ;
ear, write an equation that expresses retail price R in f terms of cost C (wholesale price). (B) What does a store pay for a suit that retails for $240? , 65. 67. 68. 69. 70. Section '32 Graphs and Lines 27 Business—Depreciation. A farmer buys a new tractor for
$157,000 and assumes that it will have a tradein value of
$82,000 after 10 years'Ihe farmer uses a constant rate of de—
preciation (commonly called straightline depreciation——
one of several methods permitted by the IRS) to determine
the annual value of the tractor. (A) Find a linear model for the depreciated value V of
the tractor t years after it was purchased. (B) What is the depreciated value of the tractor after
6 years? (C) When will the depreciated value fall below $70,000? (D) Graph V for O S t S 20 and illustrate the answers
from parts (A) and (B) on the graph. Business—Depreciation. A charter fishing company buys
a new boat for $224,000 and assumes that it will have a
trade in value of $115,200 after 16 years. (A) Find a linear model for the depreciated value V of the
boat t years after it was purchased. (B) What is the depreciated value of the tractor after
10 years? (C) When will the depreciated value fall below $100,000? (D) Graph V for 0 S t S 30 and illustrate the answers
from (A) and (B) on the graph. Boiling point. The temperature at which water starts to
boil is called its boiling point and is linearly related to the altitude. Water boils at 212°F at sea level and at 193.6°F at
an altitude of 10,000. (Sourcezbiggreenegg . com) (A) Find a relationship of the form T = mx + b where Tis
degrees Fahrenheit and x is altitude in thousands of feet. (B) Find the boiling point at an altitude at of 3.500.
(C) Find the altitude if the boiling point is 200°F. (D) Graph Tand illustrate the answers to (B) and (C) on
the graph. Boiling point. The temperature at which water starts to
boil is also linearly related to barometric pressure. Water
boils at 212°F at a pressure of 29.9 ian (inches of mer cury) and at 191°F at a pressure of 28.4 ian. (Source:
biggreenegg . com) (A) Find a relationship of the form T = mx + b, where
T is degrees Fahrenheit and x is pressure in inches of
mercury. (B) Find the boiling point at a pressure of 31 ian. (C) Find the pressure if the boiling point is 199°F. (D) Graph Tand illustrate the answers to (B) and (C) on
the graph. Flight conditions. In stable air. the air temperature drops
about 36°F for each 1,000foot rise in altitude. (Source:
Federal Aviation Administration) (A) If the temperature at sea level is 70°F, write a linear
equation that expresses temperature T in terms of
altitude A in thousands of feet. (B) At what altitude is the temperature 34°F?
Flight navigation. The airspeed indicator on some air craft is affected by the changes in atmospheric pressure
at different altitudes. A pilot can estimate the true 28 CHAPTER 1 Linear Equations and Graphs airspeed by observing the indicated airspeed and adding
to it about 1.6% for every 1,000 feet of altitude. (Source: E
i
E
i 5, TABLE 5 U.S. Barley Supply and Deriland ' 1 ‘ Ltd. Supply Demand Price
Meggmso.“ TCChfm (Ems ) . . Yen (mil bu) (mil bu) ($Ibu)
(A) If a pilot maintains a constant readmg of 200 nules per hour on the airspeed indicator as the aircraft climbs 1990 7.500 7,900 2 28
from sea level to an altitude of 10,000 feet, write a lin 1991 7,900 7,800 2 37 ear equation that expresses true airspeed T (in miles
per hour) in terms of altitudeA (in thousands of feet). (B) What would be the true airspeed of the aircraft at
6,500 feet? Source: US. Department of Agriculture 76. Supply and demand. Use the corn market data in Table 6 to
find 71. Demographics. The average number of persons per house (A) A linear supply equation 0f the form P = mx + b
hold in the Unite: Stattes has been shrinking steadily lfor as (B) A linear demand equation of the form p = m x + b. 1
long as statistics ave een kept and is approximate y lin . . . . ‘ ‘
ear with respect to time. In 1980, there were about 2.8 per (C) The eqmlrbnum pomt. _ . i .
sons per household and in 2005, about 2.6. (Source: US. (D) Gm?“ the supplyequation, demand equation. and
Census Bureau) equilibrium point in the same coordinate system. .5,
(A) If N represents the average number of persons per , household and trepresents the number ofyears since TABLE 6 US. Com Supply and Demand .‘_
1980, write a linear equation that expresses N in terms Supply Demand Price ‘
0f ‘ Year (mil bu) (mil bu) (is/bu) ;
(B) (is; 21101; :quatlon to estImate household $126 in the  1998 9,800 9,300 1.94 .5
y ' 1999 9,400 9,500 1.82 " ’ '72. Demographics The median household income divides the S m) F GE mm W k0 ,
households into two groups, the half whose income is less ome' e ' XPO e or :‘ .
than or equal to the median and the half whose income is , , _ , f ‘
greater than the median. The median household income 77' 5:331? :mke .5 law at??? tile relationsgp [9:12:69 $16
in the United States grew from about $34,000 in 1980 to ear (2 inc??? :“whic; $511112” micecznim;d)‘
about $45,000 in 2005. (Source: US. Census Bureau) For a p . plar sifting, a 51,011: d wgei gh t causes a stretch oi
(A) If 1 represents the median household income and t 2 inches, while with no weight the stretch of the spring is 0_ L. :3‘:::§£i;ﬁ:§:§;i:ﬁ:ﬁﬁnigggwme a lm (A) Find a linear equation that expresses s in terms of w. :y
(B) Use this equation to estimate median household in (B) What 1s the stretch for a weight 0f 20 pounds? '1
come in the year 2030 (C) What weight will cause a stretch of 3.6 inches? ‘
73. Cigarette smoking. The percentage of female cigarette 78 Physics. The distance d between a fixed spring and the
smokers in the United States declined from 21% in 2000 ﬂ00r is a linear function 0f the weight w attBChEd t0 the
to 18.5% in 2004. (Source: Centers for Disease Control) bottom of the Sprlng. The bottom of the spring is 18 inches i,
(A) Find a linear equation relating percentage of female from the ﬂoor when the weight is 3 pounds and 10 1nches :
smokers (f) to years since 2000 (,)_ from the ﬂoor when the weight is 5 pounds. ’ (B) Find the year in which the percentage of female (A) Find a linear equation that expresses d in terms of w. fi‘
smokers falls below 15%_ (B) Find the distance from the bottom of the spring to the  (C) Graph the equation for 0 S t s 15 and illustrate the ﬂoor If no weight 15 attached. i
answer to part (B) on the graph. (C) Find the smallest weight thatwill make the bottom of the I): _ ' t chtheﬂoor. I orethehe‘ tofth ' t. I 74. Cigarette smoking. The percentage of male Cigarette smok spring 0“ ( gn ‘gh e weigh ) :_;
ers in the United States declined from 25.7% in 2000 to 79. Energy consuMption. Table 7 lists US. oil imports/as a per it
23.4% in 2004. (Source: Centers for Disease Control) centage“ of total energy consumption for selected years i
(A) Find a linear equation relating percentage of male E smokers (m) to years since 2000 (t). ’ TABLE 7 g
(B) Find the year in which the percentage of male smok Y Oil 1m  rts % E.
ers falls below 20%. “I [m ( ) f.
(C) Graph the equation forO S t s 15 and illustrate the 1960 . _ 9 :5:
answer to part (B) on the graph. 13;: , 3 f, 75. Supply and demand. Use the barley market data in 1990 24 “
Table 5 to find 2000 29 E
(A) A linear supply equation of the form p = mx + b. Summing” Infomuﬁon Administration *3
(B) A linear demand equation of the form p = mx + b. ' j:
(C) The equilibrium pomt. ' ' * We use percentage because energy is measured in quadrillion BTUs
(D) Graph the supply equauon, demand equation, and (British Thermal Units) and one quadrillion is 1.000000000000000. E equilibrium point in the same coordinate system. Percentages are easier to comprehend. 80. Let 1 represent years since 1960 and y represent the cor responding percentage of oil imports. (A) Find the equation of the line through (D, 9) and
(40, '29). the first and last data points in Table 7. (B) Find the equation of the line through (0, 9) and (10.
12), the first and second data points in Table 7. (C) Graph the lines from parts (A) and (B) and the data : points (x, y) from Table 7 in the same coordinate
system. (D) Use each equation to predict oil imports as a per
centage of total consumption in 2020. (E) Which of the two lines seems to better represent the
data in Table 7? Discuss. Energy production. Table 8 lists US. crude oil production 7 as a percentage of total US. energy production for selected
years. Let x represent years since 1960 and y represent the
corresponding percentage of oil production. (A) Find the equation of the line through (0. 35) and
' (40, 17). the first and last data points in Table 8. Linear Regression 29 (B) Find the equation of the line through (0, 35) and
(10. 32). the ﬁrst and second data points in Table 8. (C) Graph the lines from parts (A) and (B) and the data
points (x, y) from Table 8 in the same coordinate system. (D) Use each equation to predict oil imports as a per
centage of total consumption in 2020. (E) Which of the two lines seems to better represent the
data in Table 8‘? Discuss. TABLE 8.
Year Oil Production (“/o)
.  i garﬂwwww H‘ W megawmu . w
I970 32
1980 27
N90 22
20le l 7 Source: Energy Information Administration Section 13 LINEAR REGRESSION :s Slope as a Rate of Change
as Linear Regression Mathematical modeling is the process of using mathematics to solve real—world prob
lems. This process can be broken down into three steps (Fig. 1): Step 1. Construct the mathematical model, a problem whose solution will provide
information about the realworld problem. Step 2. Solve the mathematical model. Step 3. Interpret the solution to the mathematical model in terms of the original realworld problem.
ReaLworld
problem of
s‘
.. Mathematical
solution In more complex problems, this cycle may have to be repeated several times to obtain
the required information about the realworld problem. In this section we will discuss
one of the simplest mathematical models, a linear equation. With the aid of a graphing calculator or computer, we also will learn how to analyze a linear model based on
realword data. _, Mathematical FIGURE 1 is Slope as a Rate of Change If x and y are related by the equation y = mx + b, where m and b are constants with
m ¢ 0, then x and y are linearly related. If (x1, y1) and (x2, y2) are two distinct points
on this line, then the slope of the line is 42 C H A P T E R 1 Linear Equations and Graphs for the price—supply data where x is supply (in billions of Problems 27~30 require a graphing calculator or a computer .
bushels) and y is price (in dollars). Do the same for the price thar can calculate the linear regression line for a given data set. 27' Olympic Games. Find a linear regression model for the demand data. (Round regression coefficients to two decimal
men’s IOU—meter freestyle data given in Table 16, where x
is years since 1968 and y is winning time (in seconds). Do
the same for the women’s loometer freestyle data. (Round
regression coefficients to three decimal places.) Do these places) Find the equilibrium price for corn. . TABLE 17 Supply and Demand ‘or U 5. Com models indicate that the women will eventually catch up Price .s‘fl’l‘lll Price Demand
with the men? If so, when? Do you think this will actually ($Ihu) (h'lh'm b“) ($Ibu) . (billion b“)
occur? 2.15 6.29 2.07 y ’ 9.78
2.29 7.27 2.15 9.35
. TABLE 16 Wirmlnq Tlmes 2.! wmmr ‘Sm'lmmmg Events 2.36 7.53 2.22 8.47
‘ 2.48 7.93 2.34 8.12
IOO—Meter Freestyle ZOOMeter Backstroke 2.47 8.12 2.39 7.76
M“ w"'“°“ M“ WNW“ 2.55 8.24 2.47 6.98 .
1968 5220 60'0 20960 22480 Source: www,usda.gov/nass/pubs/histdata.htm
1972 51.22 58.59 2:02.82 2:19.19
1976 49.99 55.65 1:59.19 2:13.43
13:2 32:3 22;: 28(1):: :32; 30. Supply and demand. Table 18 containspricefsupply data
' and price—demand data for soybeans. Find a linear regres
1988 48'“ 5493 1:59'37 209‘” sion model for the price—supply data where x is supply (in
1992 4902 5465 15847 2117.06 billions of bushels) and yis price (in dollars).Do the same
1996 , 4874 5450 158'54 2:0783 for the price—demand data. (Round regression coefficients
2000 4830 5383 ‘ 1:5676 20816 to two decimal places) Find the equilibrium price for soy
2004 48.17 53.84 1:54.76 2.09:16 beans Source: www.infoplease.com TABLE 18 Supply and Demand in: l.) S Swheans 28. Olympic Games. Find a linear regression model for the Price Supply Price Demand
men‘s 200meter backstroke data given in Table 16, where (Slbu) (billion bu) ($Ibu) (billion bu)
x is years since 1968 and y is winning time (in seconds). . ~ , Do the same for the women’s ZOO—meter backstroke data. 5'15 1'55 4'93 2‘60
(Round regression coefficients to four decimal places.) Do 5'79 , 1'86 ' 5'48 ' 2‘40
these models indicate that the women will eventually catch 588 1'94  571 ' 2'18
up with the men? If so, when? Do you think this will ac 607 _ 208 6m 205
tually occur? 6.15 ‘ 2.15 6.40 1.95 6.25 ' ‘i 2.27 6.66 1.85 29. Supply and demand. Table 17 contains pricesupply data and
price—demand data for corn. Find a linear regression model Source: www.usda.gov/nasslpubs/histdata.htm , CHAPTER 1 REVIEW _
:11 ”Westerns. Symkmsaevd Concepts 11 Linear Equations and Inequalities Examples ' A firstdegree, or linear, equation in one variable is any equation that can be written in the form Standard form: ax + b = 0 a at 0 ’If the equality sign in the standard form is replaced by <, >, S , or 2 , the resulting expression
is called a ﬁrstdegree, or linear, inequality. ' A solution of an equation (or inequality) involving a single variable is a number that when sub—
stituted for the variable makes the equation (inequality) true. The set of all solutions is called the solution set.
' If we perform an operation on an equation (or inequality) that produces another equation ' Ex. 1, p. 3
(or inequality)with the same...
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