Download Graphs and Functions in the Cartesian Coordinate System

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
122
(3-2)
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
(3-3)
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
124
(3-4)
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
126
(3-6)
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
(3-8)
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
(3-10)
Chapter 3
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,