Graphing a Line Given a Point and the Slope

Up to now, in this tutorial, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

Example: How To Graph a Line Given a Point and The Slope

Graph the line passing through the point \(\left(1,-1\right)\) whose slope is \(m=\frac{3}{4}\).

Solution

This table has three columns and four rows. The first row says, “Step 1. Plot the given point. Plot (1, negative 1).” To the right is a graph of the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis of the plane runs from negative 3 to 4. The point (0, negative 1) is plotted.

The second row says, “Step 2. Use the slope formula m equals rise divided by run to identify the rise and the run.” The rise and run are 3 and 4, so m equals 3 divided by 4.

The third row says “Step 3. Starting at the given point, count out the rise and run to mark the second point.” We start at (1, negative 1) and count the rise and run. Up three units and right 4 units. In the graph on the right, an additional two points are plotted: (1, 2), which is 3 units up from (1, negative 1), and (5, 2), which is 3 units up and 4 units right from (1, negative 1).

The fourth row says “Step 4. Connect the points with a line.” On the graph to the right, a line is drawn through the points (1, negative 1) and (5, 2). This line is also the hypotenuse of the right triangle formed by the three points, (1, negative 1), (1, 2) and (5, 2).

Graph a line given a point and the slope.

Plot the given point.
Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.
Starting at the given point, count out the rise and run to mark the second point.
Connect the points with a line.

Example

Graph the line with y-intercept 2 whose slope is \(m=-\frac{2}{3}\).

Solution

Plot the given point, the y-intercept, \(\left(0,2\right)\).

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (0, 2) is plotted.

\(\begin{array}{ccccc}\text{Identify the rise and the run.}\hfill & & \hfill m& =\hfill & -\frac{2}{3}\hfill \\ & & \hfill \frac{\text{rise}}{\text{run}}& =\hfill & \frac{-2}{3}\hfill \\ & & \hfill \text{rise}& =\hfill & -2\hfill \\ & & \hfill \text{run}& =\hfill & 3\hfill \end{array}\)

Count the rise and the run. Mark the second point.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The points (0, 2), (0, 0), and (3,0) are plotted and labeled. The line from (0, 2) to (0, 0) is labeled “down 2” and the line from (0, 0) to (3, 0) is labeled “right 3”.

Connect the two points with a line.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (0, 2) and (3,0).

You can check your work by finding a third point. Since the slope is \(m=-\frac{2}{3}\), it can be written as \(m=\frac{2}{-3}\). Go back to \(\left(0,2\right)\) and count out the rise, 2, and the run, \(-3\).

Example

Graph the line passing through the point \(\left(-1,-3\right)\) whose slope is \(m=4.\)

Solution

Plot the given point.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. The point (negative 1, negative 3) is plotted and labeled.

\(\begin{array}{ccccc}\text{Identify the rise and the run.}\hfill & & \hfill m& =\hfill & 4\hfill \\ \text{Write 4 as a fraction.}\hfill & & \hfill \frac{\text{rise}}{\text{run}}& =\hfill & \frac{4}{1}\hfill \\ & & \hfill \text{rise}& =\hfill & 4\phantom{\rule{0.5em}{0ex}}\text{run}=1\hfill \end{array}\)

Count the rise and run and mark the second point.

This figure shows how to graph the line passing through the point (negative 1, negative 3) whose slope is 4. The first step is to identify the rise and run. The rise is 4 and the run is 1. 4 divided by 1 is 4, so the slope is 4. Next we count the rise and run and mark the second point. To the right is a graph of the x y-coordinate plane. The x and y-axes run from negative 5 to 5. We start at the plotted point (negative 1, negative 3) and count the rise, 4. We reach the point negative 1, 1, which we plot. We then count the run from this point, which is 1. We reach the point (0, 1), which is plotted. The last step is to connect the two points with a line. We draw a line which passes through the points (negative 1, negative 3) and (0, 1).

Connect the two points with a line.

The graph shows the x y coordinate plane. The x and y-axes run from negative 5 to 5. A line passes through the plotted points (-1, -3) and (1,0).

You can check your work by finding a third point. Since the slope is \(m=4\), it can be written as \(m=\frac{-4}{-1}\). Go back to \(\left(-1,-3\right)\) and count out the rise, \(-4\), and the run, \(-1\).

This lesson is part of:

Graphs and Equations

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