3.2 Graphing by Plotting Points

Scott McDaniel

3.2 Graphing by Plotting Points

Learning Objectives

After completing this section, you should be able to:

Graph a linear equation by plotting points.
Recognize and graph vertical and horizontal lines.
Identify relationships between slope and direction of a line.
Interpret slope and intercepts from equations in different forms.

Introduction

In the previous section, we learned how to locate and name points on the coordinate plane.

Now, we will use those points to draw graphs of linear equations—straight lines that represent all solutions to an equation in two variables.

A linear equation can be written in several forms, such as:

$y = mx + b$ (slope-intercept form)
$Ax + By = C$ (standard form)

Each solution $(x,y)$ is a point that lies on the line.

Graphing by Plotting Points

To graph a linear equation, we can find several points that satisfy it and then draw the straight line through those points.

Example 3.2.1 — Graph by Plotting Points

Graph: $y = 2x + 1$

Solution

Step 1: Choose several x-values.

Step 2: Substitute into the equation to find y.

Step 3: Plot each point and draw a straight line through them.

**Table 3.2A**
x	y	Ordered Pair
$0$	$1$	$(0,1)$
$1$	$3$	$(1,3)$
$2$	$5$	$(2,5)$

The 3 Points are plotted with a line through them — **Figure 3.2A**

Try It

Example 3.2.2 — Graph from Standard Form

Graph: $2x + y = 6$

Solution

Solve for $y$ :

$y = -2x + 6$

**Table 3.2B**
x	y	Ordered Pair
$0$	$6$	$(0,6)$
$2$	$2$	$(2,2)$
$3$	$0$	$(3,0)$

The 3 points are plotted and a line goes through them. — **Figure 3.2B**

Try It

Graphing Horizontal and Vertical Lines

Some linear equations create lines that are perfectly flat (horizontal) or perfectly straight up-and-down (vertical).

Horizontal Lines

A horizontal line has the form $y = a$ , where $a$ is a constant.

All points have the same y-value.
The slope is $0$ .

Example 3.2.3 — Graph a Horizontal Line

Graph: $y = 5$

Solution

Every point on this line has $y = 5$ , such as:

$(-3,5),\; (0,5),\; (4,5)$

Shortcut: You do not need to plot points— $y = 5$ is a horizontal line that crosses the y-axis at 5.

Graph of line y = 5. The three points are plotted with a line through them. — **Figure 3.2C**

Vertical Lines

A vertical line has the form $x = a$ , where $a$ is a constant.

All points have the same x-value.
The slope is undefined.

Example 3.2.4 — Graph a Vertical Line

Graph: $x = -3$

Solution

Every point on this line has $x = -3$ , such as:

$(-3,-2),\; (-3,0),\; (-3,4)$

Shortcut: $x = -3$ is a vertical line that crosses the x-axis at $-3$ .

Graph of the three points plotted and a vertical line through them — **Figure 3.2D**

Tip

Horizontal lines have the form $y = \text{number}$ .

Vertical lines have the form $x = \text{number}$ .

Try It

Mixed Practice: Recognizing Line Types

Before graphing, identify the type of line you expect.

**Table 3.2C**
Equation Type	Example	What to Expect
Slope-intercept	$y = 3x + 2$	Diagonal line; slope 3
Standard form	$2x + y = 6$	Diagonal line; often easiest after solving for $y$
Vertical line	$x = -3$	Vertical line; undefined slope
Horizontal line	$y = 5$	Horizontal line; slope 0