Graphs with Intercepts

Identify the x- and y- Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the \(x\)- axis and the \(y\)- axis. These points are called the intercepts of the line.

Intercepts of a Line

The points where a line crosses the \(x\)- axis and the \(y\)- axis are called the intercepts of a line.

Let's look at the graphs of the lines in Figure 4.18.

Four figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negat

Figure 4.18 Examples of graphs crossing the \(x\)- negative axis.

First, notice where each of these lines crosses the \(x\) negative axis. See Figure 4.18.

Figure	*The line crosses the \(x\)-* axis at:**	Ordered pair of this point
Figure (a)	3	\((3,0)\)
Figure (b)	4	\((4,0)\)
Figure (c)	5	\((5,0)\)
Figure (d)	0	\((0,0)\)

Table 4.24

Do you see a pattern?

For each row, the \(y\)- coordinate of the point where the line crosses the \(x\)- axis is zero. The point where the line crosses the \(x\)- axis has the form (a,0) and is called the \(x\)- intercept of a line. The \(x\)- intercept occurs when y is zero.

Now, let's look at the points where these lines cross the \(y\)- axis. See Table 4.25.

Figure	The line crosses the y-axis at:	Ordered pair for this point
Figure (a)	6	\((0,6)\)
Figure (b)	−3	\((0,−3)\)
Figure (c)	−5	\((0,5)\)
Figure (d)	0	\((0,0)\)

Table 4.25

What is the pattern here?

In each row, the \(x\)- coordinate of the point where the line crosses the \(y\)- axis is zero. The point where the line crosses the \(y\)- axis has the form \((0,b\)) and is called the \(y\)- intercept of the line. The \(y\)- intercept occurs when \(x\) is zero.

\(x\)- Intercept and \(y\)- Intercept of a line

The \(x\)- intercept is the point \((a,0)\) where the line crosses the \(x\)- axis.

The \(y\)- intercept is the point \((0,b)\) where the line crosses the \(y\)- axis.

x- intercept and y- intercept of a line

Example 4.19

Find the x- and y- intercepts on each graph.

Three figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negat

Solution

The graph crosses the \(x\)- axis at the point \((4,0)\). The \(x\)- intercept is \( (4,0)\).
The graph crosses the \(y\)- axis at the point \((0,2)\). The \(y\)- intercept is \( (0,2)\).
The graph crosses the \(x\)- axis at the point \((2,0)\). The \(x\)- intercept is \((2,0)\)
The graph crosses the \(y\)- axis at the point \((0,−6)\). The \(y\)- intercept is \( (0,−6)\).
The graph crosses the \(x\)- axis at the point \((−5,0)\). The \(x\)- intercept is \((−5,0)\).
The graph crosses the \(y\)- axis at the point \((0,−5)\). The \(y\)- intercept is \( (0,−5)\).

Try It 4.37

Find the \(x\)- and \(y\)- intercepts on the graph.

A figure showing a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- a

Try It 4.38

Find the \(x\)- and \(y\)- intercepts on the graph.

The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- a