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# GRAPHING LINES

## Overview

In this lesson, you will review how to graph linear equations on a
Cartesian coordinate system . You will also find the x- and y-intercepts, the
slope of a line, and the distance between two points .

Finally, you will review how to write the equation of a line in three forms:
point-slope form, standard form, and slope- intercept form .

## Explain

Concept 1 has sections on

• The Cartesian Coordinate
System

• The Distance Formula

• Graphing a Linear Equation

• Finding x- and y-intercepts

• Horizontal and Vertical
Lines

• The Slope of a Line

CONCEPT 1:
GRAPHING LINES

The Cartesian Coordinate System

The Cartesian coordinate system consists of two real number lines
placed at right angles to each other.

The horizontal number line is called the x-axis.

The vertical number line is called the y-axis.

The axes define a flat surface called the xy-plane.

Every point in the xy-plane has two numbers associated with it.

• The x-coordinate or abscissa tells how far the point lies to the left or
right of the y-axis.

• The y-coordinate or ordinate tells how far the point lies above or below
the x-axis.

The x-coordinate and the y-coordinate are often written inside parentheses ,
like this : (x, y).

The first number, x, represents the x-coordinate and the second number,
y, represents the y-coordinate.

For example, the point that is 3 units to the right of the y-axis and 6 units
below the x-axis is labeled (3, -6).

Because the order in which the pair of numbers is written is important,
(x, y) is called an ordered pair. Thus, the point ( -6, 3) is not the same as
the point (3, -6).

The x-axis and the y-axis intersect at the point (0, 0). This point is called
the origin.

The x-axis and the y-axis divide the xy -plane into four regions called

 Quadrant Sign of x Sign of y I positive positive II negative positive III negative negative IV positive negative

A point on an axis does not lie in a quadrant.

Example EII.E.1

Find the coordinates of each point
labeled on the graph.

Then, state the quadrant in which
each point lies.

Solution

Point A has coordinates (2, 5), and lies in Quadrant I.

Point B has coordinates (3, -4), and lies in Quadrant IV.

Point C has coordinates ( -5, 3), and lies in Quadrant II.

Point D has coordinates ( -6, -2), and lies in Quadrant III.

Point E has coordinates (5, 0). It is not in a quadrant since it lies on the
x-axis.

Example EII.E.2

Plot each point on a Cartesian coordinate system:

Solution

The plot the point (5, -4), start at the origin:

• move 5 units to the right;
• then move down 4 units;
• place a dot at this location.

Follow a similar procedure for the other points. Notice the difference
between the locations of points ( -3, 0) and (0, -3).

The Distance Formula

The distance between any two points in the xy-plane can be found using
the distance formula.
 — Formula — The Distance FormulaLet (x1, y1) and (x2, y2) represent any two points in the xy-plane. The distance, d, between the points is given by If the points lie on a horizontal line, then this simplifies to If the points lie on a vertical line, then this simplifies to
In the distance formula, it doesn’t matter
which point is considered (x1, y1) or (x2, y2).
The resulting distance is the same.
Example EII.E.3

Find the distance between ( -3, 4) and (5, 4).

Solution

Since the points ( -3, 4) and (5, 4) have the same
y-coordinate, 4, they lie on a horizontal line. So,

 we use the formula: Let x1 =-3 and x2 =5. Simplify. Find the absolute value.

The distance between ( -3, 4) and (5, 4) is 8 units

In the formula, , it doesn’t
matter which x-coordinate is assigned to
x1 and which is assigned to x2. The
resulting distance is the same.

That is:

If we switch the points and let
and
, we get the

We can use a calculator to
approximate

Example EII.E.4

Find the distance between ( -3, 8) and (5, -1).

Solution
The points do not lie on a horizontal or vertical line

 Therefore, use the distance formula. Let and . Substitute. Simplify.

The distance between the points ( -3, 8) and (5, -1) is units.

Since two points determine a line, you can
use any two points that satisfy the equation
to draw the line. Plotting more than two
points will help avoid errors.
Graphing a Linear Equation

An equation that can be written in the form is called a linear
equation because its graph is a straight line.

Here are some examples:

 (Here, A 0.) (Here, B 0.)

We can graph a linear equation on a Cartesian coordinate system by
plotting points that satisfy the equation.

 — Procedure — To Graph a Linear Equation Step 1 Make a table of ordered pairs that satisfy the equation. Step 2 Plot the ordered pairs. Step 3 Draw a line through the plotted points.
Here’s how to calculate the corresponding
value for y when x = -3.
Equation.
Replace x with -3.
Multiply.
Divide both sides by -6.

Example EII.E.5

Graph the linear equation .

Solution

Step 1 Make a table of ordered pairs that satisfy the equation.

Select values for x and then use
to calculate the
corresponding values for y.

Step 2 Plot the ordered pairs.

The points are shown on the graph.

Step 3 Draw a line through the plotted points.

 x y -3 -6 0 -4 3 -2 6 0

Each point on the line represents a solution of the equation

Finding x- and y-intercepts

The point where a graph crosses an axis called an intercept.

 — Definition — x-intercept and y-intercept The x-intercept is the point where the line crosses the x-axis. The x-intercept has the form (a, 0), where a is a constant. The y-intercept is the point where the line crosses the y-axis. The y-intercept has the form (0, b), where b is a constant.
The x-intercept lies on the x-axis, so y= 0.

The y-intercept lies on the y-axis, so x= 0.
Example EII.E.6

Given the equation

a. Find the x-intercept.
b. Find the y-intercept.
c. Use the intercepts to graph the line.

Solution

a. The x-intercept has the form (a, 0).
To find the x-intercept, substitute 0 for y .
Then, solve for x .
The x-intercept is (4, 0).

b. The y-intercept has the form (0, b).
To find the y-intercept, substitute 0 for x.
Then, solve for y.
The y-intercept is (0, -2).

c. To graph the line , first plot
the x- and y-intercepts.
 x y Then, draw a line through the intercepts. 4 0 ←x-intercept As a check, it is a good idea to find a third 0 -2 ←y-intercept point on the line. -2 -3 ←check point

For example, let x = -2 in . Then, solve for y.
The result is y = -3. So, the point ( -2, -3) should also lie on the line.

Two points that are often easy to find are
the x- and y-intercepts.
Horizontal and Vertical Lines

The graph of a linear equation, , where A and B are
not both 0, is a straight line.

• If A =0, then the graph is a horizontal line. For example, y= 3 is a
horizontal line. On this line, the y-coordinate is always 3.

• If B =0, then the graph is a vertical line. For example, x =5 is a
vertical line. On this line, the x-coordinate is always 5
Example EII.E.7

Graph the following linear equations:

Solution

a. The graph of y 4 is a horizontal line.
On this line, the y-coordinate is always 4.

The y-intercept is (0, 4); the line has no x-intercept.
 x y -6 4 -4 4 0 4 7 4
b. The graph of x =4 is a vertical line.
On this line, the x-coordinate is always 4.

The x-intercept is (4, 0); the line has no y-intercept.
 x y 4 -7 4 -2 4 0 4 5
The Slope of a Line

The rise of a line represents the vertical change when moving from one
point to a second point on a line.

The run of a line represents the horizontal change when moving from one
point to a second point on the line.

The slope of a line is the ratio of the rise to the run. It is a number that
describes the steepness of the line.
 — Definition — Slope of a Line The slope of the line that passes through two points, (x1, y1) and (x2, y2), is given by slope where
Example EII.E.8

Find the slope of the line that passes through the points (-2, 7) and (4, 3).

Solution

 Let   and . Substitute these values in the slope formula. Simplify. Reduce.

Thus, the slope of the line through ( -2, 7) and (4, 3) is .

When using the slope formula, it does not
matter which point we choose for (x1, y1)
and which we choose for(x2, y2).

Example EII.E.9

Find the slope of the line that passes through the points ( -4, 5) and (3, 5).

Solution

 Let and . Substitute these values in the slope formula. Simplify. Divide.

Thus, the slope of the line through ( -4, 5) and (3, 5) is 0.
In fact, the slope of any horizontal line is 0.

Example EII.E.10

Find the slope of the line that passes through the points (3, 2) and (3, -4).

Solution

 Let and . Substitute these values in the slope formula. Simplify.

Since division by zero is undefined, the slope is undefined.
In fact, the slope of any vertical line is undefined.

 — Summary — Slope of a Lin Slope is positive: Line slants upward as we move from left to right. • Slope is negative: Line slants downward as we move from left to right. • Slope is zero: The slope of a horizontal line is 0. • Slope is undefined: The slope of a vertical line is undefined.

We can use slope to help us construct the graph of a line.

Example EII.E.11

Graph the line that passes through the point ( -5, 6) with slope
.
Solution

First, plot the given point, ( -5, 6).
To find another point on the line, use the slope. The slope,
,
tells us how to move up and down (rise) and left and right (run) to get to
another point on the line. The slope says to move 3 units
down and 4 units right to get to another point, ( -1, 3).
Plot the point ( -1, 3).
Finally, draw a line through the two points.

Here is a summary of this concept from Interactive Mathematics .

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