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Graphing with Intercepts

Identify the intercepts on a graph

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.

Let’s look at the graph of the lines shown in .

First, notice where each of these lines crosses the x- axis:

Figure:	The line crosses the x-axis at:	Ordered pair of this point
42	3	(3,0)
43	4	(4,0)
44	5	(5,0)
45	0	(0,0)

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 the line. The x- intercept occurs when y is zero.

Now, let's look at the points where these lines cross the y-axis.

Figure:	The line crosses the y-axis at:	Ordered pair for this point
42	6	(0,6)
43	-3	(0,-3)
44	-5	(0,-5)
45	0	(0,0)

Example

Try it.

Find the \(x\text{- and}\ y\text{-intercepts}\) of each line:

ⓐ \(\ x+2y=4\)
ⓑ \(\ 3x-y=6\)
ⓒ \(\ x+y=-5\)

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).

Find the Intercepts from an Equation of a Line

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

Example

Try it.

Find the intercepts of \(2x+y=6\)

Solution

We'll fill in .

To find the x- intercept, let \(y=0\):


Substitute 0 for y.
Add.
Divide by 2.
The x-intercept is (3, 0).

To find the y- intercept, let \(x=0\):


Substitute 0 for x.
Multiply.
Add.
The y-intercept is (0, 6).

The intercepts are the points \((3,0)\) and \((0,6)\).

Example

Try it.

Find the intercepts of \(4x-3y=12.\)

Solution

To find the \(x\text{-intercept,}\) let \(y=0.\)

	\(4x-3y=12\)
Substitute 0 for \(y.\)	\(4x-3\cdot 0=12\)
Multiply.	\(4x-0=12\)
Subtract.	\(4x=12\)
Divide by 4.	\(x=3\)

The \(x\text{-intercept}\) is \((3,0).\)

To find the \(y\text{-intercept},\) let \(x=0.\)

	\(4x-3y=12\)
Substitute 0 for \(x.\)	\(4\cdot 0-3y=12\)
Multiply.	\(0-3y=12\)
Simplify.	\(-3y=12\)
Divide by −3.	\(y=-4\)

The \(y\text{-intercept}\) is \((0,-4).\)

The intercepts are the points \((-3,0)\) and \((0,-4).\)

\(4x-3y=12\)
x	y
\(3\)	\(0\)
\(0\)	\(-4\)

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

Try it.

Graph \(-x+2y=6\) using intercepts.

Solution

First, find the \(x\text{-intercept}.\) Let \(y=0,\)

\(\begin{array}{l} \\ -x+2y=6 \\ -x+2(0)=6 \\ -x=6 \\ x=-6\end{array}\)

The \(x\text{-intercept}\) is \((-6,0).\)

Now find the \(y\text{-intercept}.\) Let \(x=0.\)

\(\begin{array}{l} \\ -x+2y=6 \\ -0+2y=6 \\ \\ \\ 2y=6 \\ y=3\end{array}\)

The \(y\text{-intercept}\) is \((0,3).\)

Find a third point. We’ll use \(x=2,\)

\(\begin{array}{l} \\ -x+2y=6 \\ -2+2y=6 \\ \\ \\ 2y=8 \\ y=4\end{array}\)

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.

Example

Try it.

Graph \(4x-3y=12\) using intercepts.

Solution

Find the intercepts and a third point.

We list the points and show the graph.

\(4x-3y=12\)
\(x\)	\(y\)	\((x,y)\)
\(3\)	\(0\)	\((3,0)\)
\(0\)	\(-4\)	\((0,-4)\)
\(6\)	\(4\)	\((6,4)\)

Example

Try it.

Graph \(y=5x\) using the intercepts.

Solution

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 \(-1.\)

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.

Choose the Most Convenient Method to Graph a Line

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

	Equation	Method
#1	\(y=2x+1\)	Plotting points
#2	\(y=\frac{1}{2}x+3\)	Plotting points
#3	\(x=-7\)	Vertical line
#4	\(y=4\)	Horizontal line
#5	\(2x+y=6\)	Intercepts
#6	\(4x-3y=12\)	Intercepts

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.

Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form \(Ax+By=C\). We substituted \(y=0\) and \(x=0\) to find the x- and y- intercepts, and then found a third point by choosing a value for x or y.

This leads to the following strategy for choosing the most convenient method to graph a line.

Example

Try it.

Identify the most convenient method to graph each line:

ⓐ \(\ y=-3\\)
ⓑ \(\ 4x-6y=12\\)
ⓒ \(\ x=2\\)
ⓓ \(\ y=\frac{2}{5}x-1\)

Solution

ⓐ \(\ y=-3\)

This equation has only one variable, \(y.\) Its graph is a horizontal line crossing the \(y\text{-axis}\) at \(-3.\)

ⓑ \(\ 4x-6y=12\)

This equation is of the form \(Ax+By=C.\) Find the intercepts and one more point.

There is only one variable, \(x.\) The graph is a vertical line crossing the \(x\text{-axis}\) at \(2.\)

ⓓ \(\ y=\frac{2}{5}x-1\)

Since \(y\) is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Intercepts
- The x-intercept is the point, \((a,0)\), where the graph crosses the x-axis. The x-intercept occurs when y is zero.
- The y-intercept is the point, \((0,b)\), where the graph crosses the y-axis. The y-intercept occurs when x is zero.
- The x-intercept occurs when y is zero.
- The y-intercept occurs when x is zero.
Find the x and y intercepts from the equation of a line
- To find the x-intercept of the line, let \(y=0\) and solve for x.
- To find the y-intercept of the line, let \(x=0\) and solve for y.
  x y
  0
  0
Graph a line using the intercepts
1. Find the x- and y- intercepts of the line.
  - Let \(y=0\) and solve for x.
  - Let \(x=0\) and solve for y.
2. Find a third solution to the equation.
3. Plot the three points and then check that they line up.
4. Draw the line.
Choose the most convenient method to graph a line
1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
  \(x=a\) is a vertical line passing through the x-axis at a.
  \(y=b\) is a horizontal line passing through the y-axis at b.
2. Determine if y is isolated on one side of the equation. The graph by plotting points.
  Choose any three values for x and then solve for the corresponding y- values.
3. Determine if the equation is of the form \(Ax+By=C\), find the intercepts.
  Find the x- and y- intercepts and then a third point.

Graphing with Intercepts

Identify the Intercepts on a Graph

In the following exercises, find the \(x\text{-}\) and \(y\text{-}\) intercepts.

Try it.

Solution

(3,0),(0,3)

Try it.

Solution

(5,0),(0,−5)

Try it.

Solution

(−2,0),(0,−2)

Try it.

Solution

(−1,0),(0,1)

Try it.

Solution

(0,0)

Try it.

Find the \(x\) and \(y\) Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

Try it.

\(x+y=4\)

Solution

(4,0),(0,4)

Try it.

\(x+y=3\)

Try it.

\(x+y=-2\)

Solution

(−2,0),(0,−2)

Try it.

\(x+y=-5\)

Try it.

\(x-y=5\)

Solution

(5,0),(0,−5)

Try it.

\(x-y=1\)

Try it.

\(x-y=-3\)

Solution

(−3,0),(0,3)

Try it.

\(x-y=-4\)

Try it.

\(x+2y=8\)

Solution

(8,0),(0,4)

Try it.

\(x+2y=10\)

Try it.

\(3x+y=6\)

Solution

(2,0),(0,6)

Try it.

\(3x+y=9\)

Try it.

\(x-3y=12\)

Solution

(12,0),(0,−4)

Try it.

\(x-2y=8\)

Try it.

\(4x-y=8\)

Solution

(2,0),(0,−8)

Try it.

\(5x-y=5\)

Try it.

\(2x+5y=10\)

Solution

(5,0),(0,2)

Try it.

\(2x+3y=6\)

Try it.

\(3x-2y=12\)

Solution

(4,0),(0,−6)

Try it.

\(3x-5y=30\)

Try it.

\(y=\frac{1}{3}x-1\)

Solution

(3,0),(0,−1)

Try it.

\(y=\frac{1}{4}x-1\)

Try it.

\(y=\frac{1}{5}x+2\)

Solution

(−10,0),(0,2)

Try it.

\(y=\frac{1}{3}x+4\)

Try it.

\(y=3x\)

Solution

(0,0)

Try it.

\(y=-2x\)

Try it.

\(y=-4x\)

Solution

(0,0)

Try it.

\(y=5x\)

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

Try it.

\(-x+5y=10\)

Solution

Try it.

\(-x+4y=8\)

Try it.

\(x+2y=4\)

Solution

Try it.

\(x+2y=6\)

Try it.

\(x+y=2\)

Solution

Try it.

\(x+y=5\)

Try it.

\(x+y=3\)

Solution

Try it.

\(x+y=-1\)

Try it.

\(x-y=1\)

Solution

Try it.

\(x-y=2\)

Try it.

\(x-y=-4\)

Solution

Try it.

\(x-y=-3\)

Try it.

\(4x+y=4\)

Solution

Try it.

\(3x+y=3\)

Try it.

\(3x-y=-6\)

Solution

Try it.

\(2x-y=-8\)

Try it.

\(2x+4y=12\)

Solution

Try it.

\(3x+2y=12\)

Try it.

\(3x-2y=6\)

Solution

Try it.

\(5x-2y=10\)

Try it.

\(2x-5y=-20\)

Solution

Try it.

\(3x-4y=-12\)

Try it.

\(y=-2x\)

Solution

Try it.

\(y=-4x\)

Try it.

\(y=x\)

Solution

Try it.

\(y=3x\)

Choose the Most Convenient Method to Graph a Line

In the following exercises, identify the most convenient method to graph each line.

Try it.

\(x=2\)

Solution

vertical line

Try it.

\(y=4\)

Try it.

\(y=5\)

Solution

horizontal line

Try it.

\(x=-3\)

Try it.

\(y=-3x+4\)

Solution

plotting points

Try it.

\(y=-5x+2\)

Try it.

\(x-y=5\)

Solution

intercepts

Try it.

\(x-y=1\)

Try it.

\(y=\frac{2}{3}x-1\)

Solution

plotting points

Try it.

\(y=\frac{4}{5}x-3\)

Try it.

\(y=-3\)

Solution

horizontal line

Try it.

\(y=-1\)

Try it.

\(3x-2y=-12\)

Solution

intercepts

Try it.

\(2x-5y=-10\)

Try it.

\(y=-\frac{1}{4}x+3\)

Solution

plotting points

Try it.

\(y=-\frac{1}{3}x+5\)

Try it.

How do you find the \(x\text{-intercept}\) of the graph of \(3x-2y=6?\)

Solution

Answers will vary.

Try it.

How do you find the \(y\text{-intercept}\) of the graph of \(5x-y=10?\)

Try it.

Do you prefer to graph the equation \(4x+y=-4\) by plotting points or intercepts? Why?

Solution

Answers will vary.

Try it.

Do you prefer to graph the equation \(y=\frac{2}{3}x-2\) by plotting points or intercepts? Why?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Solve: \(3x+4y=-12\) for \(x\) when \(y=0.\)
If you missed this problem, review .

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\(-4\)
Is the point \((0,-5)\) on the \(x\text{-axis}\) or \(y\text{-axis?}\)
If you missed this problem, review .

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\(y\text{-axis}\)
Which ordered pairs are solutions to the equation \(2x-y=6?\)
ⓐ \(\ (6,0)\\)ⓑ \(\ (0,-6)\\)ⓒ \(\ (4,-2).\)
If you missed this problem, review .

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b
Find the \(x\text{- and}\ y\text{-intercepts}\) of each line:
ⓐ \(\ x+2y=4\)
ⓑ \(\ 3x-y=6\)
ⓒ \(\ x+y=-5\)

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ⓐ
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).
Find the \(x\text{-}\) and \(y\text{-intercepts}\) of the graph: \(x-y=2.\)

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x-intercept (2,0): y-intercept (0,−2)
Find the \(x\text{-}\) and \(y\text{-intercepts}\) of the graph: \(2x+3y=6.\)

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x-intercept (3,0); y-intercept (0,2)
Find the intercepts of \(2x+y=6\)

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We'll fill in .
To find the x- intercept, let \(y=0\):
Substitute 0 for y.
Add.
Divide by 2.
The x-intercept is (3, 0).
To find the y- intercept, let \(x=0\):
Substitute 0 for x.
Multiply.
Add.
The y-intercept is (0, 6).
The intercepts are the points \((3,0)\) and \((0,6)\).
Find the intercepts: \(3x+y=12\)

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\((4,0)\) and \((0,12)\)
Find the intercepts: \(x+4y=8\)

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\((8,0)\) and \((0,2)\)
Find the intercepts of \(4x-3y=12.\)

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To find the \(x\text{-intercept,}\) let \(y=0.\)
\(4x-3y=12\)
Substitute 0 for \(y.\) \(4x-3\cdot 0=12\)
Multiply. \(4x-0=12\)
Subtract. \(4x=12\)
Divide by 4. \(x=3\)
The \(x\text{-intercept}\) is \((3,0).\)
To find the \(y\text{-intercept},\) let \(x=0.\)
\(4x-3y=12\)
Substitute 0 for \(x.\) \(4\cdot 0-3y=12\)
Multiply. \(0-3y=12\)
Simplify. \(-3y=12\)
Divide by −3. \(y=-4\)
The \(y\text{-intercept}\) is \((0,-4).\)
The intercepts are the points \((-3,0)\) and \((0,-4).\)
\(4x-3y=12\)
x y
\(3\) \(0\)
\(0\) \(-4\)
Find the intercepts of the line: \(3x-4y=12.\)

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x-intercept (4,0); y-intercept: (0,−3)
Find the intercepts of the line: \(2x-4y=8.\)

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x-intercept (4,0); y-intercept: (0,−2)
Graph \(-x+2y=6\) using intercepts.

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First, find the \(x\text{-intercept}.\) Let \(y=0,\)
\(\begin{array}{l} \\ -x+2y=6 \\ -x+2(0)=6 \\ -x=6 \\ x=-6\end{array}\)
The \(x\text{-intercept}\) is \((-6,0).\)
Now find the \(y\text{-intercept}.\) Let \(x=0.\)
\(\begin{array}{l} \\ -x+2y=6 \\ -0+2y=6 \\ \\ \\ 2y=6 \\ y=3\end{array}\)
The \(y\text{-intercept}\) is \((0,3).\)
Find a third point. We’ll use \(x=2,\)
\(\begin{array}{l} \\ -x+2y=6 \\ -2+2y=6 \\ \\ \\ 2y=8 \\ y=4\end{array}\)
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.
Graph the line using the intercepts: \(x-2y=4.\)

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Graph the line using the intercepts: \(-x+3y=6.\)

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Graph \(4x-3y=12\) using intercepts.

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Find the intercepts and a third point.
We list the points and show the graph.
\(4x-3y=12\)
\(x\) \(y\) \((x,y)\)
\(3\) \(0\) \((3,0)\)
\(0\) \(-4\) \((0,-4)\)
\(6\) \(4\) \((6,4)\)
Graph the line using the intercepts: \(5x-2y=10.\)

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Graph the line using the intercepts: \(3x-4y=12.\)

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Graph \(y=5x\) using the intercepts.

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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 \(-1.\)

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.
Graph using the intercepts: \(y=4x.\)

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Graph using the intercepts: \(y=-x.\)

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Identify the most convenient method to graph each line:
1. ⓐ \(\ y=-3\\)
2. ⓑ \(\ 4x-6y=12\\)
3. ⓒ \(\ x=2\\)
4. ⓓ \(\ y=\frac{2}{5}x-1\)
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ⓐ \(\ y=-3\)
This equation has only one variable, \(y.\) Its graph is a horizontal line crossing the \(y\text{-axis}\) at \(-3.\)
ⓑ \(\ 4x-6y=12\)
This equation is of the form \(Ax+By=C.\) Find the intercepts and one more point.
ⓒ \(\ x=2\)
There is only one variable, \(x.\) The graph is a vertical line crossing the \(x\text{-axis}\) at \(2.\)
ⓓ \(\ y=\frac{2}{5}x-1\)
Since \(y\) is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points.
Identify the most convenient method to graph each line:
1. ⓐ \(\ 3x+2y=12\)
2. ⓑ \(\ y=4\)
3. ⓒ \(\ y=\frac{1}{5}x-4\)
4. ⓓ \(\ x=-7\)
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1. ⓐ intercepts
2. ⓑ horizontal line
3. ⓒ plotting points
4. ⓓ vertical line
Identify the most convenient method to graph each line:
1. ⓐ \(\ x=6\)
2. ⓑ \(\ y=-\frac{3}{4}x+1\)
3. ⓒ \(\ y=-8\)
4. ⓓ \(\ 4x-3y=-1\)
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1. ⓐ vertical line
2. ⓑ plotting points
3. ⓒ horizontal line
4. ⓓ intercepts
\(2x+5y=10\)

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(5,0),(0,2)
\(3x-2y=12\)

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(4,0),(0,−6)
\(3x-5y=30\)
\(y=\frac{1}{3}x-1\)

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(3,0),(0,−1)
\(y=\frac{1}{4}x-1\)
\(y=\frac{1}{5}x+2\)

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(−10,0),(0,2)
\(y=\frac{1}{3}x+4\)
\(-x+5y=10\)

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\(2x+4y=12\)
\(3x+2y=12\)
\(5x-2y=10\)
\(2x-5y=-20\)

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\(3x-4y=-12\)
\(y=\frac{2}{3}x-1\)

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plotting points
\(y=\frac{4}{5}x-3\)
\(3x-2y=-12\)

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intercepts

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Graphing with Intercepts

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
the
the
Find the
Let

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

بیشتر در Arithmetic

Order of operations Fractions Percentages Powers and roots