Mathematics » Introducing Graphs » Understand Slope of a Line

Graphing a Line Given a Point and the Slope

Graphing a Line Given a Point and the Slope

In this tutorial, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.

Example

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

Solution

Plot the given point, \(\left(1,-1\right).\)

 

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 7. The y-axis runs from -3 to 4. A labeled point is drawn at “ordered pair 1, -1”.

Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.

\(\begin{array}{}\\ \\ \phantom{\rule{0.8em}{0ex}}m=\frac{3}{4}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{3}{4}\hfill \\ \\ \\ \phantom{\rule{0.2em}{0ex}}\text{rise}=3\hfill \\ \phantom{\rule{0.33em}{0ex}}\text{run}=4\hfill \end{array}\)

Starting at the point we plotted, count out the rise and run to mark the second point. We count \(3\) units up and \(4\) units right.

 

The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two line segments are drawn. A vertical line segment connects the points “ordered pair 1, -1” and “order pair “1, 2”. It is labeled “3”. A horizontal line segment starts at the top of the vertical line segment and goes to the right, connecting the points “ordered pair 1, 2” and “ordered pair 5, 2”. It is labeled “4”.

Then we connect the points with a line and draw arrows at the ends to show it continues.

The graph shows the x y-coordinate plane. The x-axis runs from -3 to 5. The y-axis runs from -1 to 7. Two unlabeled points are drawn at  “ordered pair 1, -1” and  “ordered pair 5, 2”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 1, -1” and “ordered pair 1, 2 ”.  A horizontal line segment connects “ordered pair 1, 2” and “ordered pair 5, 2”.

We can check our line by starting at any point and counting up \(3\) and to the right \(4.\) We should get to another point on the line.

How to Graph a line given a point and a slope.

  1. Plot the given point.
  2. Use the slope formula to identify the rise and the run.
  3. Starting at the given point, count out the rise and run to mark the second point.
  4. Connect the points with a line.

Example

Graph the line with \(y\)-intercept \(\left(0,2\right)\) and slope \(m=-\frac{2}{3}.\)

Solution

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

The graph shows the x y-coordinate plane. The x-axis runs from -1 to 4. The y-axis runs from -1 to 3. The point “ordered pair 0, 2” is labeled.

Use the slope formula \(m=\frac{\text{rise}}{\text{run}}\) to identify the rise and the run.

\(\begin{array}{}\\ \\ \phantom{\rule{0.8em}{0ex}}m=-\frac{2}{3}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{-2}{3}\hfill \\ \\ \\ \phantom{\rule{0.2em}{0ex}}\text{rise}=–2\hfill \\ \phantom{\rule{0.3em}{0ex}}\text{run}=3\hfill \end{array}\)

Starting at \(\left(0,2\right),\) count the rise and the run and mark the second point.

The graph shows the x y-coordinate plane. Both axes run from -5 to 5. A vertical line segment connects points at “ordered pair 0, 2” and “ordered pair 0, 0” and is labeled “down 2”. A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 0, 3” and is labeled “right 3”.

Connect the points with a line.

The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at  “ordered pair 0, 2” and  “ordered pair 3, 0”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair 0, 2” and “ordered pair 0, 0 ”.  A horizontal line segment connects “ordered pair 0, 0” and “ordered pair 3, 0”.

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. Both axes run from -5 to 5. The point “ordered pair -1, -3” is labeled.

Identify the rise and the run.\(m=4\)
Write 4 as a fraction.\(\frac{\text{rise}}{\text{run}}=\frac{4}{1}\)
 \(\text{rise}=4\phantom{\rule{0.2em}{0ex}}\text{run}=1\)

Count the rise and run.

The graph shows the x y-coordinate plane. Both axes run from -5 to 5. The y-axis runs from -4 to 2. A vertical line segment connects points at “ordered pair -1,  -3” and “ordered pair -1, 1” and is labeled “up 4”. A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1” and is labeled “over 1”.

Mark the second point. Connect the two points with a line.

The graph shows the x y-coordinate plane. Both axes run from -5 to 5. Two labeled points are drawn at  “ordered pair -1, -3” and  “ordered pair -1, 1”.  A line passes through the points. Two line segments form a triangle with the line. A vertical line connects “ordered pair -1, -3” and “ordered pair -1, 1 ”. It is labeled “up 4” A horizontal line segment connects “ordered pair -1, 1” and “ordered pair 0, 1”. It is labeled “over 1”

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