1. Copyright © Cengage Learning. All rights reserved.
Graphs and Functions
Copyright © Cengage Learning. All rights reserved.

2. 4.1 Ordered Pairs and Graphs

3. What You Will Learn Plot points on a rectangular coordinate system
Determine whether ordered pairs are solutions of equations
Use the verbal problem-solving method to plot points on a rectangular coordinate system

4. The Rectangular Coordinate System

5. The Rectangular Coordinate System
Just as you can represent real numbers by points on the real number line, you can represent ordered pairs of real numbers by points in a plane.
This plane is called a rectangular coordinate system or the Cartesian plane, after the French mathematician René Descartes (1596–1650).

6. The Rectangular Coordinate System
A rectangular coordinate system is formed by two real lines intersecting at right angles.

7. The Rectangular Coordinate System
The horizontal number line is usually called the x-axis and the vertical number line is usually called the y-axis. (The plural of axis is axes.).
The point of intersection of the two axes is called the origin, and the axes separate the plane into four regions called quadrants.
Each point in the plane corresponds to an ordered pair (x, y) of real numbers x and y, called the coordinates of the point.

8. Example 1 – Plotting points on a Rectangular Coordinate System
Plot the points (–1, 2), (3, 0), (2, –1), (3, 4), (0, 0), and (–2, –3) on a rectangular coordinate system.
Solution:
The point (–1, 2) is one unit to the left of the vertical axis and two units above the horizontal axis.

9. Example 1 – Plotting points on a Rectangular Coordinate System
cont’d
Similarly, the point (3, 0) is three units to the right of the vertical axis and on the horizontal axis. (It is on the horizontal axis because the y-coordinate is zero.)
The other four points can be plotted in a similar way, as shown in Figure 4.3.

10. Example 2 – Graphing Super Bowl Scores
The scores of the Super Bowl games from
1992 through 2012 are in the table. Plot these
points on a rectangular coordinate system.
(Source: National Football League)

11. Example 2 – Graphing Super Bowl Scores
cont’d
Solution:
The x-coordinate of the points represents the year, and the y-coordinate represents the winning and losing scores. The winning scores are shown as black dots, and the losing scores are shown as blue dots. Note that the break in the x-axis indicates tat the numbers between 0 and 1992 have been omitted.

12. Ordered Pairs as Solutions of Equations

13. Ordered Pairs as Solutions of Equations
In mathematics, the relationship between the variables
x and y is often given by an equation.
From the equation, you can construct your own table of values. For instance, consider the equation
y = 2x + 1.
To construct a table of values for this equation, choose several x-values and then calculate the corresponding y-values.

14. Ordered Pairs as Solutions of Equations
For example, if you choose x = 1, the corresponding y-value is
y = 2(1) + 1
y = 3.
The corresponding ordered pair (x, y) = (1, 3) is a solution point (or solution) of the equation.
Substitute 1 for x.
Simplify.

15. Ordered Pairs as Solutions of Equations
The table below is a table of values (and the corresponding solution points) using x-values of –3, –2, –1, 0, 1, 2, and 3.
These x-values are arbitrary. You should try to use x-values that are convenient and simple to use.

16. Ordered Pairs as Solutions of Equations
Once you have constructed a table of values, you can get a visual idea of the relationship between the variables x and y by plotting the solution points on a rectangular coordinate system.
For instance, the solution points shown in the table.

17. Ordered Pairs as Solutions of Equations
In many places throughout this course, you will see that approaching a problem in different ways can help you understand the problem better.
For instance, the discussion above looks at solutions of an equation in three ways.

18. Example 3 – Constructing a Table of Values
Construct a table of values showing five solution points for the equation
6x – 2y = 4.
Then plot the solution points on a rectangular coordinate system. Choose x-values of –2, –1, 0, 1, and 2.
Solution:
6x – 2y = 4
6x – 6x – 2y = 4 – 6x
–2y = –6x + 4
Write original equation.
Subtract 6x from each side.
Combine like terms.

19. Example 3 – Constructing a Table of Values
cont’d
y = 3x – 2
Now, using the equation y = 3x – 2, you can construct a table of values, as shown below.
Divide each side by –2.
Simplify.

20. Example 3 – Constructing a Table of Values
cont’d
Finally, from the table you can plot the five solution points on a rectangular coordinate system.

21. Ordered Pairs as Solutions of Equations
In the next example, you are given several ordered pairs and are asked to determine whether they are solutions of the original equation.
To do this, you need to substitute the values of x and y into the equation.
If the substitution produces a true statement, the ordered pair (x, y) is a solution and is said to satisfy the equation.

22. Ordered Pairs as Solutions of Equations

23. Example 4 – Verifying Solutions of an Equations
Determine whether each of the ordered pairs is a solution of x + 3y = 6.
a. (1, 2) b. (0, 2)
Solution:
a. For the ordered pair (1, 2), substitute x = 1 and y = 2 into the original equation.
x + 3y = 6
1 + 3(2) 6
7  6
Because the substitution does not satisfy the original equation, you can conclude that the ordered pair (1, 2) is not a solution of the original equation.
Write original equation.
Substitute 1 for x and 2 for y.
Not a solution

24. Example 4 – Verifying Solutions of an Equations
cont’d
b. For the ordered pair (0, 2), substitute x = 0 and y = into the original equation.
x + 3y = 6
0 + 3(2) 6
6 = 6
Because the substitution satisfies the original equation, you can conclude that the ordered pair (0, 2) is a solution of the original equation.
Write original equation.
Substitute 0 for x and 2 for y.
Solution

25. Applications

26. Example 5 – Finding the Total Cost
You set up a small business to assemble computer keyboards. Your initial cost is $120,000, and your unit cost of assembling each keyboard is $40. Write an equation that relates your total cost to the number of keyboards produced. Then plot the total cost of producing 1000, 2000, 3000, 4000, and 5000 keyboards.
Solution:
Verbal
Model:
Labels: Total cost = c (dollars)
Unit cost = (dollars per keyboard)
Number of keyboards = x (keyboards)
Initial cost = 120, (dollars)

27. Example 5 – Finding the Total Cost
cont’d
Expression: C = 40x + 120,000
Using this equation, you can construct the following table of values.
From the table, you can plot the ordered pairs.
Although graphs can help you visualize
relationships between two variables,
they can also be misleading, as shown
in the next example.

28. Example 6 – Identify Misleading Graphs
The graphs shown below represent the yearly profits for a truck rental company. Which graph is misleading? Why?

29. Example 6 – Identify Misleading Graphs
cont’d
Solution:
a. This graph is misleading. The scale on the vertical axis makes it appear the change in profits from 2009 to 2013 is dramatic, but the total change is only $3000, which is small in comparison with $3,000,000.
b. This graph is truthful. By showing the full scale on the y-axis, you can see that, relative to the overall size of the profit, there was almost no change from one year to the next.