4.3 Graph with Intercepts

Learning objectives

Learning objectives

• 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 the 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.

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

First, notice where each of these lines crosses the $x$-axis. See Figure 4.18.

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.

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.

Example 4.19

Solution

1. ⓐ 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 $(0,2)$.

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 $(0,-6)$.

3. ⓒ 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 $(0,-5)$.

Find the x- and y- Intercepts from an Equation of a Line

Recognizing that the x- intercept occurs when y is zero and that the y- intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x- intercept, let $y=0$ and solve for x. To find the y- intercept, let $x=0$ and solve for y.

Example 4.20

Solution

We will let $y=0$ to find the x- intercept, and let $x=0$ to find the y- intercept. We will fill in the table, which reminds us of what we need to find.

To find the x- intercept, let $y=0$.

The intercepts are the points $(3,0)$ and $(0,6)$ as shown in Table 4.26.

Table

4.26

Example 4.21

Solution

The intercepts are the points (3, 0) and (0, −4) as shown in the following table.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

Example 4.22

The steps to graph a linear equation using the intercepts are summarized below.

Example 4.23

Solution

Find the intercepts and a third point.

We list the points in Table 4.27 and show the graph below.

Table

4.27

Example 4.24

Solution

This line has only one intercept. It is the point $(0,0)$.

To ensure accuracy we need to plot three points. Since the x- and y- intercepts are the same point, we need two more points to graph the line.

See Table 4.28.

Table

4.28

Plot the three points, check that they line up, and draw the line.

Section 4.3 Exercises

Section 4.3 Exercises

Identify the x- and y- Intercepts on a Graph

In the following exercises, find the x- and y- intercepts on each graph.

Find the x- and y- Intercepts from an Equation of a Line

In the following exercises, find the intercepts for each equation.

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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

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

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

1. Find the intercepts of $x+4y=8.$

2. Find the intercepts of $3x-4y=12.$

3. Find the intercepts of $2x-4y=8.$

1. Graph $x-2y=4$ using the intercepts.

2. Graph $-x+3y=6$ using the intercepts.

3. Graph $5x-2y=10$ using the intercepts.

1. Graph $3x-4y=12$ using the intercepts.

2. Graph $y=4x$ using the intercepts.

3. Graph $y=-x$ the intercepts.

1. $x+y=4$

2. $x+y=-2$

3. $x-y=5$

1. $x-y=-3$

2. $x+2y=8$

3. $3x+y=6$

1. $x-3y=12$

2. $4x-y=8$

3. $2x+5y=10$

1. $3x-2y=12$

2. $y=\frac{1}{3}x+1$

3. $y=\frac{1}{5}x+2$

1. $y=3x$

2. $y=-4x$

3. $-x+5y=10$

1. $x+2y=4$

2. $x+y=2$

3. $x+y=-3$

1. $x-y=1$

2. $x-y=-4$

3. $4x+y=4$

1. $2x+4y=12$

2. $3x-2y=6$

3. $2x-5y=-20$

1. $3x-y=-6$

2. $y=-2x$

3. $y=x$

1. Road trip. Damien is driving from Chicago to Denver, a distance of 1000 miles. The x- axis on the graph below shows the time in hours since Damien left Chicago. The y- axis represents the distance he has left to drive.

1. ⓐ Find the x- and y- intercepts.

2. ⓑ Explain what the x- and y- intercepts mean for Damien.

2. How do you find the x- intercept of the graph of $3x-2y=6$?

3. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=\frac{2}{3}x-2$? Why?