Graphing a Line Given a Point and a Slope

Learning Outcomes

  • Graph a line given the slope and a point on the line
  • Solve slope applications

In this chapter, 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 [latex]\left(1,-1\right)[/latex] whose slope is [latex]m=\Large\frac{3}{4}[/latex].

Solution
Plot the given point, [latex]\left(1,-1\right)[/latex].

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
Use the slope formula [latex]m=\Large\frac{\text{rise}}{\text{run}}[/latex] to identify the rise and the run.

[latex]\begin{array}{}\\ \\ m=\frac{3}{4}\hfill \\ \frac{\text{rise}}{\text{run}}=\frac{3}{4}\hfill \\ \\ \\ \text{rise}=3\hfill \\ \text{run}=4\hfill \end{array}[/latex]

Starting at the point we plotted, count out the rise and run to mark the second point. We count [latex]3[/latex] units up and [latex]4[/latex] 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
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
We can check our line by starting at any point and counting up [latex]3[/latex] and to the right [latex]4[/latex]. We should get to another point on the line.

 

try it

 

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 [latex]y[/latex] -intercept [latex]\left(0,2\right)[/latex] and slope [latex]m=-\Large\frac{2}{3}[/latex]

 

try it

 

example

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

 

try it

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

example

The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

This figure shows a house with a sloped roof. The roof on one half of the building is labeled

 

try it

 

example

Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.

Sewage pipes must slope down [latex]{\Large\frac{1}{4}}[/latex] inch per foot in order to drain properly. What is the required slope?

This figure shows a right triangle. The short leg is vertical and is labeled

 

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In the following video we show another example of how to graph a line given a point on the line and the slope of the line.