Mathematics » Introducing Graphs » Graphing with Intercepts

Graphing a Line Using the Intercepts

Graphing 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

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

Solution

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

\(\begin{array}{}\\ \phantom{\rule{0.7em}{0ex}}-x+2y=6\\ -x+2\left(0\right)=6\\ \phantom{\rule{2.8em}{0ex}}-x=6\\ \phantom{\rule{4.3em}{0ex}}x=-6\end{array}\)

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

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

\(\begin{array}{}\\ -x+2y=6\\ -0+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=6\\ \phantom{\rule{3em}{0ex}}y=3\end{array}\)

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

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

\(\begin{array}{}\\ -x+2y=6\\ -2+2y=6\\ \\ \\ \phantom{\rule{2.4em}{0ex}}2y=8\\ \phantom{\rule{3em}{0ex}}y=4\end{array}\)

A third solution to the equation is \(\left(2,4\right).\)

Summarize the three points in a table and then plot them on a graph.

\(-x+2y=6\)
xy(x,y)
\(-6\)\(0\)\(\left(-6,0\right)\)
\(0\)\(3\)\(\left(0,3\right)\)
\(2\)\(4\)\(\left(2,4\right)\)

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.

Do the points line up? Yes, so draw line through the points.

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10. Three labeled points are shown at “ordered pair -6, 0”, “ordered pair 0, 3” and “ordered pair 2, 4”.  A line passes through the three labeled points.

How to Graph a line using the intercepts.

  1. Find the \(x-\) and \(\text{y-intercepts}\) 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.

Example

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

Solution

Find the intercepts and a third point.

The figure shows 3 solutions to the equation 4 x - 3 y = 12. The first is titled “x-intercept, let y = 0”. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for y, reading 4 x - 3 open parentheses 0 closed parentheses = 12. The third line is 4 x = 12. The last line is x = 3. The second solution is titled “y-intercept, let x = 0”. The first line is 4 x - 3 y = 12. The second line shows 0 in red substituted for x, reading 4 open parentheses 0 closed parentheses - 3 y = 12. The third line is -3 y = 12. The last line is y = -4. The third solution is titled “third point, let y = 4”. The first line is 4 x - 3 y = 12. The second line shows 4 in red substituted for y, reading 4 x - 3 open parentheses 4 closed parentheses = 12. The third line is 4 x - 12 = 12. The last line is x = 6.

We list the points and show the graph.

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

The graph shows the x y-coordinate plane. Both axes run from -7 to 7. Three unlabeled points are drawn at  “ordered pair 0, -4”, “ordered pair 3, 0” and “ordered pair  6, 4”.  A line passes through the points.

Example

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

Solution

The figure shows 2 solutions to y = 5 x. The first solution is titled “x-intercept; Let y = 0.” The first line is y = 5 x. The second line is 0, shown in red, = 5 x. The third line is 0 = x. The fourth line is x = 0. The last line is “The x-intercept is “ordered pair 0, 0”. The second  solution is titled “y-intercept; Let x = 0.” The first line is y = 5 x. The second line is  y = 5 open parentheses 0, shown in red, closed parentheses. The third line is y = 0. The last line is “The y-intercept is “ordered pair 0, 0”.

This line has only one intercept! It is the point \(\left(0,0\right).\)

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 \(-1.\)

 

The figure shows two substitutions in the equation y = 5 x. In the first substitution,  the first line is y = 5 x. The second line is y = 5 open parentheses 1, shown in red, closed parentheses. The third line is y =5. The last line is “ordered pair 1, 5”.  In the second substitution,  the first line is y = 5 x. The second line is y = 5 open parentheses -1, shown in red, closed parentheses. The third line is y = -5. The last line is “ordered pair -1, -5”.

Organize the points in a table.

\(y=5x\)
\(x\)\(y\)\(\left(x,y\right)\)
\(0\)\(0\)\(\left(0,0\right)\)
\(1\)\(5\)\(\left(1,5\right)\)
\(-1\)\(-5\)\(\left(-1,-5\right)\)

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

The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through three labeled points, “ordered pair -1, -5”, “ordered pair 0, 0”, and ordered pair 1, 5”.

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