11.5: Graphing with Intercepts (Part 1)
 Page ID
 5042
Skills to Develop
 Identify the intercepts on a graph
 Find the intercepts from an equation of a line
 Graph a line using the intercepts
 Choose the most convenient method to graph a line
be prepared!
Before you get started, take this readiness quiz.
 Solve: 3x + 4y = −12 for x when y = 0. If you missed this problem, review Example 9.11.6.
 Is the point (0, −5) on the xaxis or yaxis? If you missed this problem, review Example 11.1.5.
 Which ordered pairs are solutions to the equation 2x − y = 6? (a) (6, 0) (b) (0, −6) (c) (4, −2). If you missed this problem, review Example 11.2.8.
Identify the Intercepts on a Graph
Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.
At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line.
Definition: Intercepts of a Line
Each of the points at which a line crosses the xaxis and the yaxis is called an intercept of the line.
Let’s look at the graph of the lines shown in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\)
First, notice where each of these lines crosses the x axis:
Figure:  The line crosses the xaxis at:  Ordered pair of this point 

Figure \(\PageIndex{1a}\)  3  (3,0) 
Figure \(\PageIndex{1b}\)  4  (4,0) 
Figure \(\PageIndex{1c}\)  5  (5,0) 
Figure \(\PageIndex{1d}\)  0  (0,0) 
Do you see a pattern?
For each row, the ycoordinate of the point where the line crosses the xaxis is zero. The point where the line crosses the xaxis has the form (a, 0); and is called the xintercept of the line. The xintercept occurs when y is zero.
Now, let's look at the points where these lines cross the yaxis.
Figure:  The line crosses the xaxis at:  Ordered pair of this point 

Figure \(\PageIndex{1a}\)  6  (0, 6) 
Figure \(\PageIndex{1b}\)  3  (0, 3) 
Figure \(\PageIndex{1c}\)  5  (0, 5) 
Figure \(\PageIndex{1d}\)  0  (0, 0) 
Definition: xintercept and yintercept of a line
The xintercept is the point, (a, 0), where the graph crosses the xaxis.
The xintercept occurs when y is zero.
The yintercept is the point, (0, b), where the graph crosses the yaxis.
The yintercept occurs when x is zero.
Example \(\PageIndex{1}\)
Find the x and yintercepts of each line:
(a) x + 2y = 4
(b) 3x  y = 6
(c) x + y = 5
Solution
(a)
The graph crosses the xaxis at the point (4, 0).  The xintercept is (4, 0). 
The graph crosses the yaxis at the point (0, 2).  The xintercept is (0, 2). 
(b)
The graph crosses the xaxis at the point (2, 0).  The xintercept is (2, 0). 
The graph crosses the yaxis at the point (0, 6).  The xintercept is (0, 6). 
(c)
The graph crosses the xaxis at the point (5, 0).  The xintercept is (5, 0). 
The graph crosses the yaxis at the point (0, 5).  The xintercept is (0, 5). 
Exercise \(\PageIndex{1A}\)
Find the x and yintercepts of the graph: x − y = 2.
 Answer

xintercept (2,0); yintercept (0,2)
Exercise \(\PageIndex{1B}\)
Find the x and yintercepts of the graph: 2x + 3y = 6.
 Answer

xintercept (3,0); yintercept (0,2)
Find the Intercepts from an Equation of a Line
Recognizing that the xintercept occurs when y is zero and that the yintercept occurs when x is zero gives us a method to find the intercepts of a line from its equation. To find the xintercept, let y = 0 and solve for x. To find the yintercept, let x = 0 and solve for y.
Definition: Find the x and y from the Equation of a Line
Use the equation to find:
 the xintercept of the line, let y = 0 and solve for x.
 the yintercept of the line, let x = 0 and solve for y
x  y 

0  
0 
Example \(\PageIndex{2}\)
Find the intercepts of 2x + y = 6
Solution
We'll fill in Figure \(\PageIndex{2}\).
Figure \(\PageIndex{2}\)
To find the x intercept, let y = 0:
Substitute 0 for y.  \(2x + \textcolor{red}{0} = 6\) 
Add.  2x = 6 
Divide by 2.  x = 3 
The xintercept is (3, 0).
To find the y intercept, let x = 0:
Substitute 0 for x.  \(2 \cdot \textcolor{red}{0} + y = 6\) 
Multiply.  0 + y = 6 
Add.  y = 6 
The yintercept is (0, 6).
2x + y = 6  

x  y 
3  0 
0  6 
Figure \(\PageIndex{3}\)
The intercepts are the points (3, 0) and (0, 6).
Exercise \(\PageIndex{2A}\)
Find the intercepts: 3x + y = 12.
 Answer

xintercept (4,0); yintercept (0,12)
Exercise \(\PageIndex{2B}\)
Find the intercepts: x + 4y = 8.
 Answer

xintercept (8,0); yintercept (0,2)
Example \(\PageIndex{3}\)
Find the intercepts of 4x−3y = 12.
Solution
To find the xintercept, let y = 0.
Substitute 0 for y.  4x − 3 • 0 = 12 
Multiply.  4x − 0 = 12 
Subtract.  4x = 12 
Divide by 4.  x = 3 
The yintercept is (0, −4). The intercepts are the points (−3, 0) and (0, −4).
4x  3y = 12  

x  y 
3  0 
0  4 
Exercise \(\PageIndex{3A}\)
Find the intercepts of the line: 3x−4y = 12.
 Answer

xintercept (4,0); yintercept (0,3)
Exercise \(\PageIndex{3B}\)
Find the intercepts of the line: 2x−4y = 8.
 Answer

xintercept (4,0); yintercept (0,2)
Graph a Line Using the Intercepts
To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.
Example \(\PageIndex{4}\)
Graph −x + 2y = 6 using intercepts.
Solution
First, find the xintercept. Let y = 0,
$$\begin{split} x + 2y &= 6 \\ x + 2(0) &= 6 \\ x &= 6 \\ x &= 6 \end{split}$$
The xintercept is (–6, 0).
Now find the yintercept. Let x = 0.
$$\begin{split} x + 2y &= 6 \\ 0 + 2y &= 6 \\ 2y &= 6 \\ y &= 3 \end{split}$$
The yintercept is (0, 3).
Find a third point. We’ll use x = 2,
$$\begin{split} x + 2y &= 6 \\ 2 + 2y &= 6 \\ 2y &= 8 \\ y &= 4 \end{split}$$
A third solution to the equation is (2, 4).
Summarize the three points in a table and then plot them on a graph.
x + 2y = 6  

x  y  (x,y) 
6  0  (−6, 0) 
0  3  (0, 3) 
2  4  (2, 4) 
Do the points line up? Yes, so draw line through the points.
Exercise \(\PageIndex{4A}\)
Graph the line using the intercepts: x−2y = 4.
 Answer
Exercise \(\PageIndex{4B}\)
Graph the line using the intercepts: −x + 3y = 6.
 Answer
HOW TO: GRAPH A LINE USING THE INTERCEPTS
Step 1. Find the x  and yintercepts of the line.
 Let y = 0 and solve for x.
 Let x = 0 and solve for y.
Step 2. Find a third solution to the equation.
Step 3. Plot the three points and then check that they line up.
Step 4. Draw the line.
Example \(\PageIndex{5}\)
Graph 4x−3y = 12 using intercepts.
Solution
Find the intercepts and a third point.
$$\begin{split} xintercept,\; &let\; y = 0 \\ 4x  3y &= 12 \\ 4x  3(\textcolor{red}{0}) &= 12 \\ 4x &= 12 \\ x &= 3 \end{split}$$  $$\begin{split} yintercept,\; &let\; x = 0 \\ 4x  3y &= 12 \\ 4(\textcolor{red}{0})  3y &= 12 \\ 4x  3(\textcolor{red}{4}) &= 12 \\ 3y &= 12 \\ y &= 4 \end{split}$$  $$\begin{split} third\; point,\; &let\; y = 4 \\ 4x  3y &= 12 \\ 4x  12 &= 12 \\ 4x &= 24 \\ x &= 6 \end{split}$$ 
We list the points and show the graph.
4x  3y = 12  

x  y  (x. y) 
3  0  (3, 0) 
0  4  (0, −4) 
6  4  (6, 4) 
Exercise \(\PageIndex{5A}\)
Graph the line using the intercepts: 5x−2y = 10.
 Answer
Exercise \(\PageIndex{5B}\)
Graph the line using the intercepts: 3x−4y = 12.
 Answer
Example \(\PageIndex{6}\)
Graph \(y = 5x\) using the intercepts.
Solution
$$\begin{split} xintercept;\; &Let\; y = 0 \ldotp \\ y &= 5x\\ \textcolor{red}{0} &= 5x \\ 0 &= x \\ x &= 0 \\ The\; xintercept\; &is\; (0, 0) \ldotp \end{split}$$  $$\begin{split} yintercept;\; &Let\; x = 0 \ldotp \\ y &= 5x \\ y &= 5(\textcolor{red}{0}) \\ y &= 0 \\ The\; yintercept\; &is\; (0, 0) \ldotp \end{split}$$ 
This line has only one intercept! It is the point (0, 0).
To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for x, so we’ll let x be 1 and −1.
$$\begin{split} x &= 1 \\ y &= 5x \\ y &= 5(\textcolor{red}{1}) \\ y &= 5 \\ (1, &5) \end{split}$$  $$\begin{split} x &= 1 \\ y &= 5x \\ y &= 5(\textcolor{red}{1}) \\ y &= 5 \\ (1, &5) \end{split}$$ 
Organize the points in a table.
y = 5x  

x  y  (x, y) 
0  0  (0, 0) 
1  5  (1, 5) 
1  5  (−1, −5) 
Plot the three points, check that they line up, and draw the line.
Exercise \(\PageIndex{6A}\)
Graph using the intercepts: \(y = 3x\).
 Answer
Exercise \(\PageIndex{6B}\)
Graph using the intercepts: \(y = − x\).
 Answer
Contributors
Lynn Marecek (Santa Ana College) and MaryAnne AnthonySmith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1fa2...49835c3c@5.191."