3.3 Graphing Using Intercepts

Scott McDaniel

3.3 Graphing Using Intercepts

Learning Objectives

After completing this section, you should be able to:

Identify the x- and y-intercepts on a graph.
Find the x- and y-intercepts from an equation of a line.
Graph a line using only its intercepts.

Introduction

In the last section, you learned to graph a line by plotting several points.

While that method always works, it can be time-consuming.

In this section, you will learn a faster method: graphing using intercepts.

Since most lines cross both axes, we can use just those two points—the intercepts—to draw the entire line.

Understanding Intercepts

The x-intercept is where the line crosses the x-axis. At this point, $y = 0$ .
The y-intercept is where the line crosses the y-axis. At this point, $x = 0$ .

Example 3.3.1

Look at the graph of a line and identify the intercepts.

Linear graph with intercepts (0,-6) and (3,0) — **Figure 3.3A**

Solution

The line crosses the x-axis at $(3,0)$ .

The line crosses the y-axis at $(0,-6)$ .

x-intercept: $(3,0)$

y-intercept: $(0,-6)$

Try It

Finding Intercepts from an Equation

You can find intercepts algebraically by setting one variable equal to zero and solving for the other.

Example 3.3.2

Find the intercepts of $2x + 3y = 12$ .

Solution

Step 1: Find the x-intercept

Set $y = 0$ .

$2x + 3(0) = 12$

$2x = 12$

$x = 6$

x-intercept: $(6,0)$

Step 2: Find the y-intercept

Set $x = 0$ .

$2(0) + 3y = 12$

$3y = 12$

$y = 4$

y-intercept: $(0,4)$

Here is the graph:

Graph with intercepts (0,4) and (6,0). — **Figure 3.3B**

Example 3.3.3

Find the intercepts of $5x + 5y = 20$ .

Solution

Set $y = 0$ :

$5x = 20$

$x = 4$

x-intercept: $(4,0)$

Set $x = 0$ :

$5y = 20$

$y = 4$

y-intercept: $(0,4)$

Linear Graph with intercepts (0,4) and (4, 0) — **Figure 3.3C**

Example 3.3.4

Find the intercepts of $y = 2x - 6$ .

Solution

x-intercept: set $y = 0$

$0 = 2x - 6$

$2x = 6$

$x = 3$

x-intercept: $(3,0)$

y-intercept: set $x = 0$

$y = -6$

y-intercept: $(0,-6)$

Graph with line through (0, -6) and (3,0) — **Figure 3.3D**

Try It

Graphing a Line Using Intercepts

Once you have both intercepts:

Plot the x-intercept on the x-axis.
Plot the y-intercept on the y-axis.
Draw a straight line through the two points.

Every straight line is determined by two distinct points.

Example 3.3.5

Graph $2x + 3y = 12$ using intercepts.

Solution

x-intercept: $(6,0)$

y-intercept: $(0,4)$

Linear Graph with intercepts (6,0) and (0,4) — **Figure 3.3E**

Try It

Example 3.3.6

Graph $y = -x - 4$ using intercepts.

Solution

Set $y = 0$ : $0 = -x - 4$ → $x = -4$ → $(-4,0)$

Set $x = 0$ : $y = -4$ → $(0,-4)$

Linear graph with intercepts (-4,0) and (0,-4) — **Figure 3.3F**

Try It

Special Cases

Some lines do not cross both axes.

• Horizontal lines (such as $y = 4$ ) cross the y-axis but never the x-axis.

• Vertical lines (such as $x = -2$ ) cross the x-axis but never the y-axis.

Example 3.3.7

Graph $y = 4$ .

Solution

This is a horizontal line through $(0,4)$ .

Linear graph y = 4. A line through the y-intercept of 4 — **Figure 3.3G**

Example 3.3.8

Graph $x = -2$ .

Solution

This is a vertical line through $(-2,0)$ .

Graph of vertical line x = -2 — **Figure 3.3H**

Tip

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

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

Summary Table

**Table 3.3**
Equation	x-intercept	y-intercept	Line Type
$2x + 3y = 12$	$(6,0)$	$(0,4)$	Diagonal
$5x + 5y = 20$	$(4,0)$	$(0,4)$	Diagonal
$y = 4$	None	$(0,4)$	Horizontal
$x = -2$	$(-2,0)$	None	Vertical

Key Takeaways

To find the x-intercept, set $y = 0$ .
To find the y-intercept, set $x = 0$ .
Two intercepts are enough to graph any non-vertical line.
Intercepts provide a fast way to graph without building a full table.

License

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Introduction

Understanding Intercepts

Finding Intercepts from an Equation

Graphing a Line Using Intercepts

Special Cases

Tip

License

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