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Graphing a Line Using Its Slope and Intercept

Graphing a Line Using Its Slope and Intercept

Now that we know how to find the slope and y-intercept of a line from its equation, we can graph the line by plotting the y-intercept and then using the slope to find another point.

Example: How to Graph a Line Using its Slope and Intercept

Graph the line of the equation \(y=4x-2\) using its slope and y-intercept.

Solution

The figure shows the steps to graph the equation y equals 4x minus 2. Step 1 is to find the slope intercept form of the equation. The equation is already in slope intercept form.Step 2 is to identify the slope and y-intercept. Use the equation y equals m x, plus b. The equation y equals m x, plus b is shown with the variable m colored red and the variable b colored blue. Below that is the equation y equals 4 x, plus -2. The number 4 is colored red and -2 is colored blue. From this equation we can see that m equals 4 and b equals -2 so the slope is 4 and the y-intercept is the point (0, negative 2).Step 3 is to plot the y-intercept. An x y-coordinate plane is shown with the x-axis of the plane running from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The point (0, negative 2) is plotted.Step 4 is to use the slope formula m equals rise over run to identify the rise and the run. Since m equals 4, rise over run equals 4 over 1. From this we can determine that the rise is 4 and the run is 1.Step 5 is to start at they-intercept, count out the rise and run to mark the second point. So start at the point (0, negative 2) and count the rise and the run. The rise is up 4 and the run is right 1. On the x y-coordinate plane is a red vertical line starts at the point (0, negative 2) and rises 4 units at its end a red horizontal line runs 1 unit to end at the point (1, 2). The point (1, 2) is plotted.Step 6 is to connect the points with a line. On the x y-coordinate plane the points (0, negative 2) and (1, 2) are plotted and a line runs through the two points. The line is the graph of y equals 4 x, minus 2.

Graph a line using its slope and y-intercept.

  1. Find the slope-intercept form of the equation of the line.
  2. Identify the slope and y-intercept.
  3. Plot the y-intercept.
  4. Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
  5. Starting at the y-intercept, count out the rise and run to mark the second point.
  6. Connect the points with a line.

Example

Graph the line of the equation \(y=\text{−}x+4\) using its slope and y-intercept.

Solution

\(y=mx+b\)
The equation is in slope–intercept form. \(y=\text{−}x+4\)
Identify the slope and y-intercept. \(m=-1\)
y-intercept is (0, 4)
Plot the y-intercept. See graph below.
Identify the rise and the run. \(m=\frac{-1}{1}\)
Count out the rise and run to mark the second point. rise −1, run 1
Draw the line. .
To check your work, you can find another point on the line and make sure it is a solution of the equation. In the graph we see the line goes through (4, 0).
Check.
\(\begin{array}{}\\ \hfill y& =\hfill & \text{−}x+4\hfill \\ \hfill 0& \stackrel{?}{=}\hfill & -4+4\hfill \\ \hfill 0& =\hfill & 0✓\hfill \end{array}\)

Example

Graph the line of the equation \(y=-\frac{2}{3}x-3\) using its slope and y-intercept.

Solution

\(y=mx+b\)
The equation is in slope–intercept form. \(y=-\frac{2}{3}x-3\)
Identify the slope and y-intercept. \(m=-\frac{2}{3}\); y-intercept is (0, −3)
Plot the y-intercept. See graph below.
Identify the rise and the run.
Count out the rise and run to mark the second point.
Draw the line. .

Example

Graph the line of the equation \(4x-3y=12\) using its slope and y-intercept.

Solution

\(4x-3y=12\)
Find the slope–intercept form of the equation. \(\phantom{\rule{1.43em}{0ex}}-3y=-4x+12\)
\(\phantom{\rule{1.26em}{0ex}}-\frac{3y}{3}=\frac{-4x+12}{-3}\)
The equation is now in slope–intercept form. \(\phantom{\rule{2.63em}{0ex}}y=\frac{4}{3}x-4\)
Identify the slope and y-intercept. \(\phantom{\rule{2.37em}{0ex}}m=\frac{4}{3}\)
y-intercept is (0, −4)
Plot the y-intercept. See graph below.
Identify the rise and the run; count out the rise and run to mark the second point.
Draw the line. .

We have used a grid with \(x\) and \(y\) both going from about \(-10\) to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.

Example

Graph the line of the equation \(y=0.2x+45\) using its slope and y-intercept.

Solution

We’ll use a grid with the axes going from about \(-80\) to 80.

\(y=mx+b\)
The equation is in slope–intercept form. \(y=0.2x+45\)
Identify the slope and y-intercept. \(m=0.2\)
The y-intercept is (0, 45)
Plot the y-intercept. See graph below.
Count out the rise and run to mark the second point. The slope is \(m=0.2\); in fraction form this means \(m=\frac{2}{10}\). Given the scale of our graph, it would be easier to use the equivalent fraction \(m=\frac{10}{50}\).
Draw the line. .

Now that we have graphed lines by using the slope and y-intercept, let’s summarize all the methods we have used to graph lines. See the figure below.

The table has two rows and four columns. The first row spans all four columns and is a header row. The header is “Methods to Graph Lines”. The second row is made up of four columns. The first column is labeled “Plotting Points” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The rest of the table is blank. Below the table it reads “Find three points. Plot the points, make sure they line up, then draw the line.” The Second column is labeled “Slope–Intercept” and shows the equation y equals m x, plus b. Below the equation it reads “Find the slope and y-intercept. Start at the y-intercept, then count the slope to get a second point.” The third column is labeled “Intercepts” and shows a smaller table with four rows and two columns. The first row is a header row with the first column labeled “x” and the second labeled “y”. The second row has a 0 in the “x” column and the “y” column is blank. The second row is blank in the “x” column and has a 0 in the “y” column. The third row is blank. Below the table it reads “Find the intercepts and a third point. Plot the points, make sure they line up, then draw the line.” The fourth column is labeled “Recognize Vertical and Horizontal Lines”. Below that it reads “The equation has only one variable.” The equation x equals a is a vertical line and the equation y equals b is a horizontal line.

This lesson is part of:

Graphs and Equations

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