C

H

A

P

T

E

R

3

Graphs and Functions in the Cartesian Coordinate System he first self-propelled automobile to carry passengers was built in 1801 by the British inventor Richard Trevithick. By 1911 about 600,000 automobiles were operated in the United States alone. Some were powered by steam and some by electricity, but most were powered by gasoline. In 1913, to meet the ever growing demand, Henry Ford increased production by introducing a moving assembly line to carry automobile parts. Today the United States is a nation of cars. Over 11 million automobiles are produced here annually, and total car registrations number over 114 million. Henry Ford’s early model T sold for $850. In 1994 the average price of a new car was $18,000, with some selling for more than $50,000. Prices for new cars rise every year, but, unfortunately for the buyer, the moment a new automobile is bought, its value begins to decrease. Much of the behavior of automobile prices can be modeled by linear equations. In Exercises 53 and 54 of Section 3.1 you will use linear equations to find increasing new car prices and depreciating used car prices.

T

List price (in thousands of dollars)

30 (2001, 27,532) 25 (1993, 20,115) 20 15 (1985, 12,674) 10 5 0

85 87 89 91 93 95 97 99 01 Year

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Chapter 3

Graphs and Functions in the Cartesian Coordinate System

3.1 In this section ●

Ordered Pairs

●

Plotting Points

●

Graphing a Linear Equation

●

Using Intercepts for Graphing

●

Applications

●

Function Notation

E X A M P L E

helpful

GRAPHING LINES IN THE COORDINATE PLANE

In Chapter 1 we graphed numbers on a number line. In Chapter 2 we used number lines to illustrate the solution sets to inequalities. In this section we graph pairs of numbers in a coordinate system made from two number lines to illustrate solution sets.

Ordered Pairs An equation in two variables, such as y  2x  3, is satisfied only if we find a value for x and a value for y that make it true. For example, if x  4 and y  11, then the equation becomes 11  2  4  3, which is a true statement. We write x  4 and y  11 as the ordered pair (4, 11). The order of the numbers in an ordered pair is important. For example, (5, 13) also satisfies y  2x  3 because using 5 for x and 13 for y gives 13  2  5  3. However, (13, 5) does not satisfy this equation because 5  2  13  3. In an ordered pair the value for x, the x-coordinate, is always written first and the value for y, the y-coordinate, is second: (x, y).

1

hint

In this chapter you will be doing a lot of graphing. Using graph paper will help you to understand the concepts and to recognize errors. For your convenience, a page of graph paper can be found at the end of this chapter. Make as many copies of it as you wish.

Writing ordered pairs Complete the following ordered pairs so that each ordered pair satisfies the equation 4x  y  5. a) (2, ) b) ( , 3)

Solution a) Replace x with 2 in 4x  y  5 because the x-coordinate is 2: 4(2)  y  5 8  y  5 y  13 So the y-coordinate is 13 and the ordered pair is (2, 13). b) Replace y with 3 in 4x  y  5 because the y-coordinate is 3: 4x  3  5 4x  2 1 x   2 So the x-coordinate is 1 and the ordered pair is 1, 3. 2

2

■

Plotting Points To graph ordered pairs of real numbers, we need a new coordinate system. The rectangular or Cartesian coordinate system consists of a horizontal number line, the x-axis, and a vertical number line, the y-axis, as shown in Fig. 3.1. The intersection of the axes is the origin. The axes divide the coordinate plane, or the xy-plane, into four regions called quadrants. The quadrants are numbered as shown in Fig. 3.1, and they do not include any points on the axes.

3.1

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Graphing Lines in the Coordinate Plane

123

y-axis

Quadrant II

6 5 4

Quadrant I

3 2 1

–6 –5 –4 –3 –2 –1 –1 –2 Quadrant III

y

Origin 1

–3 –4 –5 –6

2

3

4

5

6

x-axis

(–3, 4)

Origin

Quadrant IV

–4 –3 –2 –1 –1

FIGURE 3.1

(2, 5)

5 4 3 2 1 1 2

3

4

x

FIGURE 3.2

Just as every real number corresponds to a point on the number line, every pair of real numbers corresponds to a point in the rectangular coordinate system. For example, the pair (2, 5) corresponds to the point that lies two units to the right of the origin and five units up. See Fig. 3.2. To locate (3, 4), start at the origin and move three units to the left and four units upward, as in Fig. 3.2. Locating a point in the rectangular coordinate system is referred to as plotting or graphing the point.

E X A M P L E

2

Graphing ordered pairs Plot the points (2, 3), (2, 3), (1, 3), (0, 4), and (4, 2).

y

(–1, 3)

4 3 2 1

– 4 –3 –2 –1 –1 –2 (–2, –3) –3

Solution To plot (2, 3), start at the origin, move two units to the right, then up three units. To graph (2, 3), start at the origin, move two units to the left, then down three ■ units. All five points are shown in Fig. 3.3.

(2, 3)

1

2

3

x

4

(4, – 2)

The solution set to an equation in two variables consists of all ordered pairs that satisfy the equation. For example, the solution set to y  2x  3 can be written in set notation as (x, y)  y  2x  3. However, this set notation does not shed any light on the solution set to y  2x  3. We can get a better understanding of the solution set with a visual image or graph of the solution set.

(0, –4)

FIGURE 3.3

E X A M P L E

study

Graphing a Linear Equation

3

tip

It is a good idea to work with others, but don’t be misled. Working a problem with help is not the same as working a problem on your own. In the end, mathematics is personal. Make sure that you can do it.

Graphing a linear equation Graph the solution set to y  2x  3.

Solution If we arbitrarily choose x  4, then y is determined by the equation y  2x  3: y  2(4)  3  5 So the ordered pair (4, 5) satisfies the equation. In this manner we can make the following table of ordered pairs: x

4

3

2

1

0

1

y  2x  3

5

3

1

1

3

5

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Chapter 3

helpful

Graphs and Functions in the Cartesian Coordinate System y

Plot these ordered pairs as shown in Fig. 3.4. Of course, there are infinitely many ordered pairs that satisfy y  2x  3, but they all lie along the line in Fig. 3.4. The arrows on the ends of the line indicate that it extends without bound in both directions. The line in Fig. 3.4 is a graph of ■ the solution set to y  2x  3.

hint

The graph of a linear equation is a straight line that exists in our minds. The straight line in our minds has no thickness, is perfectly straight, and extends infinitely. All attempts to draw it on paper fall short. The best we can do is to use a sharp pencil to keep it thin, a ruler to make it straight, and arrows to indicate that it is infinite.

In an equation such as y  2x  3 the value of y depends on the value of x. So x is the independent variable and y is the dependent variable. Because the graph of y  2x  3 is a line, the equation is a linear equation.

5 4 3 y = 2x + 3

1

– 5 – 4 – 3 –2 – 1 –1 –2 –3 –4 –5

1

2

3

x

FIGURE 3.4

E X A M P L E

4

Graphing a linear equation Graph y  2x  1. Plot at least four points.

Solution If we write y in terms of x, we get y  2x  1. Now arbitrarily select four values for x and calculate the corresponding values for y:

y 4 3 1 –3 –2 –1 –1 –2 –3 –4

1 2 3

4

x

5

x

1

0

1

2

y  2x  1

3

1

1

3

■

Plot these points and draw a line through them as shown in Fig. 3.5.

y = –2x + 1

FIGURE 3.5

If the coefficient of a variable in a linear equation is 0, then that variable is usually omitted from the equation. For example, the equation y  0  x  2 is written as y  2. Because x is multiplied by 0, any value of x can be used as long as y is 2. Because the y-coordinates are all the same, the graph is a horizontal line.

calculator close-up To graph y  2x  1 with a graphing calculator, first press Y and enter y1  2x  1.

Xmin  10, X max  10, Xscl  1, Ymin  10, Ymax  10, Yscl  1 These settings are referred to as the standard window.

Press GRAPH to draw the graph. Even though the calculator does not draw a very good straight line, it supports our conclusion that Fig. 3.5 is the graph of y  2x  1.

10

10 Next press WINDOW to set the viewing window as follows:

10

10

3.1

E X A M P L E

5

Graphing Lines in the Coordinate Plane

(3-5)

125

Graphing a horizontal line Graph y  2. Plot at least four points.

y

Solution The following table gives four points that satisfy y  2, or y  0  x  2. Note that it is easy to determine y in this case because y is always 2.

5 4 3

y=2

1 –3 –2 –1 –1

1 2

3

4

x

5

E X A M P L E

6

0

1

y0x2

2

2

2

2

■

The horizontal line through these points is shown in Fig. 3.6.

Solution We can think of the equation x  4 as x  4  0  y. Because y is multiplied by 0, the equation is satisfied by any ordered pair with an x-coordinate of 4.

x=4

1

1

Graphing a vertical line Graph x  4. Plot at least four points.

y

–2 –1 –1 –2 –3

2

If the coefficient of y is 0 in a linear equation, then the graph is a vertical line.

FIGURE 3.6

4 3 2 1

x

2 3

x

5

x40y

4

4

4

4

y

2

1

0

1

The vertical line through these points is shown in Fig. 3.7.

FIGURE 3.7

■

Using Intercepts for Graphing The x-intercept of a line is the point where the line crosses the x-axis. The x-intercept has a y-coordinate of 0. Similarly, the y-intercept of a line is the point where the line crosses the y-axis. The y-intercept has an x-coordinate of 0. If a line has distinct x- and y-intercepts, then these intercepts can be used as two points that determine the location of the line. (Horizontal lines, vertical lines, and lines through the origin do not have two distinct intercepts.)

E X A M P L E

helpful

7

hint

You can find the intercepts for 3x  4y  6 using the coverup method. Cover up 4y with your pencil, then solve 3x  6 mentally to get x  2 and an x-intercept of (2, 0). Now cover up 3x and solve 4y  6 to get y  32 and a y-intercept of (0, 32).

Using intercepts to graph Use the intercepts to graph the line 3x  4y  6.

Solution Let x  0 in 3x  4y  6 to find the y-intercept: 3(0)  4y  6 4y  6 3 y   2 Let y  0 in 3x  4y  6 to find the x-intercept: 3x  4(0)  6 3x  6 x2

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Chapter 3

y 5 4 3 2 1 –3 –2 –1 –1

3

The y-intercept is 0, 2, and the x-intercept is (2, 0). The line through the intercepts is shown in Fig. 3.8. To check, find another point that satisfies the equation. ■ The point (2, 3) satisfies the equation and is on the line in Fig. 3.8.

3x – 4y = 6

Even though two points determine the location of a line, CAUTION finding at least three points will help you to avoid errors.

(2, 0) 1

2

3

x

4 5

3 (0, – — 2)

FIGURE 3.8

E X A M P L E

Applications In applications we often use variables such as C for cost, R for revenue, and n for the number of items so that it is easier to remember what the variables represent. In this case we rename the axes. Which axis is labeled with which variable is somewhat arbitrary. However, when one variable depends on, or is determined by another, the dependent variable is usually on the vertical axis and the independent variable is on the horizontal axis.

–3 –4 –5

Cost (in thousands of dollars)

Graphs and Functions in the Cartesian Coordinate System

8

Graphing a linear equation in an application The cost per week C (in dollars) of producing n pairs of shoes for the Reebop Shoe Company is given by the linear equation C  2n  8000. Graph the equation for n between 0 and 800 inclusive (0  n  800).

Solution Make a table of values for n and C as follows:

C 10

n

0

200

400

600

800

9

C  2n  8000

8000

8400

8800

9200

9600

8

200 400 600 800 Number of pairs

FIGURE 3.9

WARM-UPS

n

Graph the line as shown in Fig. 3.9. Because C  2n  8000 expresses C in terms of n, C is the dependent variable and the vertical axis is labeled C. To accommodate the large numbers, we let each unit on the n-axis represent 200 pairs and each ■ unit on the C-axis represent $1,000. Note how the axes in Fig. 3.9 are labeled. The n-axis starts at 0 and each unit represents 200 pairs. Because all of the costs were between $8,000 and $9,600 we omitted the tick marks for 1 through 7 and put a wave in the C-axis to indicate that some numbers are missing. Omitting the numbers from 1 through 7 makes the difference between the $8,000 and $9,600 costs look greater.

True or false? Explain your answer.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

The point (2, 5) satisfies the equation 3y  2x  4. False The vertical axis is usually called the x-axis. False The point (0, 0) is in quadrant I. False The point (0, 1) is on the y-axis. True The graph of x  7 is a vertical line. True The graph of 8  y  0 is a horizontal line. True The y-intercept for the line y  2x  3 is (0, 3). True If C  3n  4, then C  10 when n  2. True If P  3x and P  12, then x  36. False The vertical axis should be A when graphing A  r2. True

3.1

3. 1

(3-7)

Graphing Lines in the Coordinate Plane

EXERCISES

Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What is the point called at the intersection of the x- and y-axis? The origin is the point where the x-axis and y-axis intersect. 2. What is an ordered pair? An ordered pair is a pair of real numbers in which there is a first number and a second one. 3. What are the x- and y-intercepts? Intercepts are points where a graph crosses the axes. 4. What type of equation has a graph that is a horizontal line? The graph of an equation of the type y  k where k is a fixed number is a horizontal line. 5. What type of equation has a graph that is a vertical line? The graph of an equation of the type x  k where k is a fixed number is a vertical line. 6. Which variable usually goes on the vertical axis? The dependent variable usually goes on the vertical axis.

 

3 21. , 0 2

23. (0, 1)

x-axis y-axis

22. (3, 2)

I

24. (4, 3)



27. y  2x  3

28. y  2x  3

29. y  x

30. y  x

31. y  3

32. y  2



 

Plot the following points in a rectangular coordinate system. For each point, name the quadrant in which it lies or the axis on which it lies. See Example 2. 11. (2, 5) I 12. (5, 1) II 1 13. 3,  III 14. (2, 6) III 2 15. (0, 4) y-axis 16. (0, 2) y-axis



17. (, 1)

IV

19. (4, 3)

II

IV

Graph each linear equation. Plot four points for each line. See Examples 3–6. 25. y  x  1 26. y  x  1

Complete the given ordered pairs so that each ordered pair satisfies the given equation. See Example 1. 7. (2, ), ( , 3), y  3x  6 (2, 0), (3, 3) 1 3 8. (1, ), ( , 4), y   x  2 1,  , (4, 4) 2 2 1 1 9. (4, ), ( , 6),  x   y  9 (4, 33), (22, 6) 2 3 1 10. (3, ), ( , 1), 2x  3y  5 3,  , (1, 1) 3



127

 

4 18. , 0 x-axis 3 20. (0, 3) y-axis

128

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Chapter 3

Graphs and Functions in the Cartesian Coordinate System

33. y  1  x

34. y  2  x

43. y  0.26x  3.86

44. y  1.35x  4.27

35. x  2

36. x  3

1 37. y  x  1 2

1 38. y  x  2 3

47. x  y  5  0

48. x  y  7  0

39. x  4  0

40. y  3  0

49. 2x  3y  5

50. 3x  4y  7

41. 3x  y  5

42. x  2y  4

3 2 51. y  x   5 3

2 5 52. y  x   3 4

Find the x- and y-intercepts for each line and use them to graph the line. See Example 7. 45. 4x  3y  12 46. 2x  5y  20

3.1

Graphing Lines in the Coordinate Plane

129

applicant’s ability to repay, a higher rating indicating higher risk. The interest rate, r, is then determined by the formula r  0.02t  0.15. If your rating were 8, then what would be your interest rate? Sketch the graph of the equation for t ranging from 0 to 10. 31%

Solve the following. See Example 8. 53. Camaro inflation. The rising list price P (in dollars) for a new Camaro Z28 Coupe can be modeled by the equation P  19,663  269n, where n is the number of years since 1995 (Edmund’s New Car Prices, www.edmunds.com). a) What will be the list price for a new Z28 in 2005? b) What is the annual increase in list price? c) Sketch a graph of this equation for 0  n  10. a) $22,353 b) $269 c)

54. Camaro Z28 depreciation. The 1998 average retail price P (in dollars) for an n-year-old Camaro Z28 Coupe can be modeled by the equation P  18,675  1,960n, where 1  n  4 (Edmund’s Used Car Prices, www. edmunds.com). a) What was the average retail price of a 4-year-old Z28 in 1998? b) How much does this model depreciate each year? c) Sketch a graph of this equation for 1  n  4. a) $10,835 b) $1960 c)

(3-9)

57. Little Chicago Pizza. The equation C  0.50t  8.95 gives the customer’s cost in dollars for a pan pizza, where t is the number of toppings. a) Find the cost of a five-topping pizza. b) Find t if C  14.45 and interpret your result. a) $11.45 is the cost of a five-topping pizza. b) 11 is the number of toppings on a $14.45 pizza. 58. Long distance charges. The formula L  0.10n  4.95 gives the monthly bill in dollars for AT&T’s one rate plan, where n is the number of minutes of long distance used during the month. Answer each question and interpret the result. a) Find the bill for 120 minutes. $16.95 b) Find n if L  23.45. 185 c) What are the coordinates for the L-intercept shown on the accompanying graph? (0, 4.95) d) What are the coordinates of the n-intercept? (49.5, 0) AT&T’s One Rate Plan

Monthly bill (in dollars)

25

55. Rental cost. For a one-day car rental the X-press Car Company charges C dollars, where C is determined by the formula C  0.26m  42 and m is the number of miles driven. What is the charge for a car driven 400 miles? Sketch a graph of the equation for m ranging from 0 to 1000. $146

20 15 10 5 0

0

50 100 150 Time (minutes)

200

FIGURE FOR EXERCISE 58 59. Cost, Revenue, and Profit. Hillary sells roses at a busy Los Angeles intersection. The formulas C  0.55x  50, R  1.50x, and P  0.95x  50

56. Measuring risk. The Friendly Bob Loan Company gives each applicant a rating, t, from 0 to 10 according to the

give her weekly cost, revenue, and profit in terms of x, where x is the number of roses that she sells in one week. a) Find C, R, and P if x  850. Interpret your results. b) Find x if P  995 and interpret your result.

130

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Graphs and Functions in the Cartesian Coordinate System

c) Find R  C if x  1100 and interpret your result. a) Her weekly cost, revenue, and profit are $517.50, $1,275, and $757.50. b) 1,100. She had a profit of $995 on selling 1,100 roses. c) 995. The difference between revenue and cost is $995, which is her profit. 60. Velocity of a pop up. A pop up off the bat of Mark McGwire goes straight into the air at 88 feet per second (ft/sec). The formula v  32t  88 gives the velocity of the ball in feet per second, t seconds after the ball is hit. a) Find the velocity for t  2 and t  3 seconds. What does a negative velocity mean? b) For what value of t is v  0? Where is the ball at this time? c) What are the two intercepts on the accompanying graph? Interpret this answer. d) If the ball takes the same time going up as it does coming down, then what is its velocity as it hits the ground? a) 24 ft/sec, 8 ft/sec, going down b) 2.75 seconds, at maximum height c) (0, 88) indicates that at t  0 second the velocity was 88 ft/sec, (2.75, 0) indicates that at t  2.75 seconds the velocity was 0 ft/sec d) 88 ft/sec

c) According to a theorem in geometry, the three medians of any triangle intersect at a single point. Estimate the point of intersection of the three medians in your triangle. a) (3, 4), (4, 0), (1, 2) c) 8, 2 3

GR APHING C ALCUL ATOR EXERCISES Graph each linear equation on a graphing calculator. Choose a viewing window that shows both intercepts. Answers may vary. 63. y  3x  20

64. y  5  50x

Velocity (feet/second)

100 50 0

1

2

4

5

6

–50

65. y  300x  2

–100 Time (seconds)

FIGURE FOR EXERCISE 60 GET TING MORE INVOLVED 61. Midpoint formula. The point M midway between (x1, y1) and (x2, y2) can be found using the formula x1  x2 y1  y2 ,  . M  2 2 Find the midpoint for each of the following pairs of points and plot the points. a) (1, 2) and (5, 6) (3, 4) b) (6, 5) and (2, 1) (2, 3) 5 3 1 1 1 1 c) ,  and ,  ,  12 8 2 4 3 2 62. Intersecting medians. Using graph paper, draw a triangle with vertices (0, 6), (6, 2), and (2, 2). a) Find the midpoint of each side of your triangle. b) A median is a line segment connecting a vertex of a triangle with the midpoint of the opposite side. Draw the three medians for your triangle.



 

  





66. y  5x  800

3.2

67. x  2y  600

Slope of a Line

(3-11)

131

68. 3x  2y  1500

3.2 In this section ●

Slope

●

Using Coordinates to Find Slope

SLOPE OF A LINE

In Section 3.1 we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail and study the concept of slope of a line.

Slope

●

Parallel Lines

●

Perpendicular Lines

●

Applications of Slope

If a highway has a 6% grade, then in 100 feet (measured horizontally) the road rises 6 feet (measured vertically). See Fig. 3.10. The ratio of 6 to 100 is 6%. If a roof rises 9 feet in a horizontal distance (or run) of 12 feet, then the roof has a 9–12 pitch. A roof with a 9–12 pitch is steeper than a roof with a 6–12 pitch. The grade of a road and the pitch of a roof are measurements of steepness. In each case the measurement is a ratio of rise (vertical change) to run (horizontal change).

6%

9 ft rise

GRADE 6 100 SLOW VEHICLES KEEP RIGHT

helpful

9–12 pitch

hint

Since the amount of run is arbitrary, we can choose the run to be 1. In this case rise

slope    rise. 1

So the slope is the amount of change in y for a change of 1 in the x-coordinate.This is why rates like 50 miles per hour (mph), 8 hours per day, and two people per car are all slopes.

12 ft run

FIGURE 3.10

We measure the steepness of a line in the same way that we measure steepness of a road or a roof. The slope of a line is the ratio of the change in y-coordinate, or the rise, to the change in x-coordinate, or the run, between two points on the line. Slope

rise change in y-coordinate Slope     change in x-coordinate run Consider the line in Fig. 3.11(a) on the next page. In going from (0, 1) to (1, 3), there is a change of 1 in the x-coordinate and a change of 2 in the y-coordinate,