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4.1 Ordered Pairs and Graphs

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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

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The Rectangular Coordinate System

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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).

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The Rectangular Coordinate System

A rectangular coordinate system is formed by two real lines

intersecting at right angles.

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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.

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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.

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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.

cont’d

Example 1 – Plotting points on a Rectangular Coordinate System

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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)

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Example 2 – Graphing Super Bowl Scores

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.

cont’d

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Ordered Pairs as

Solutions of Equations

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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.

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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.

Simplify.

Substitute 1 for x.

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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.

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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.

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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.

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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

Subtract 6x from each side.

Write original equation.

Combine like terms.

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Example 3 – Constructing a Table of Values

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.

cont’d

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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.

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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.

Ordered Pairs as Solutions of Equations

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Ordered Pairs as Solutions of Equations

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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

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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.

Substitute 1 for x and 2 for y.

Write original equation.

Not a solution

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b. For the ordered pair (0, 2), substitute x = 0 and y = 2

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.

Substitute 0 for x and 2 for y.

Write original equation.

Solution

cont’d

Example 4 – Verifying Solutions of an Equations

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Applications

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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 = 40 (dollars per keyboard)

Number of keyboards = x (keyboards)

Initial cost = 120,000 (dollars)

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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.

cont’d

Example 5 – Finding the Total Cost

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Example 6 – Identify Misleading Graphs

The graphs shown below represent the yearly profits for a

truck rental company. Which graph is misleading? Why?

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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.

cont’d

Example 6 – Identify Misleading Graphs

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Homework: