Finding the Slope of a Line From Its Graph

Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.

example

Find the slope of the line shown: Solution Locate two points on the graph, choosing points whose coordinates are integers. We will use [latex]\left(0,-3\right)[/latex] and [latex]\left(5,1\right)[/latex]. Starting with the point on the left, [latex]\left(0,-3\right)[/latex], sketch a right triangle, going from the first point to the second point, [latex]\left(5,1\right)[/latex].

Count the rise on the vertical leg of the triangle.	The rise is [latex]4[/latex] units.
Count the run on the horizontal leg.	The run is [latex]5[/latex] units.
Use the slope formula.	[latex]m=\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values of the rise and run.	[latex]m=\frac{4}{5}[/latex]
The slope of the line is [latex]\frac{4}{5}[/latex] .

example

Find the slope of the line shown:

Answer: Solution Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use [latex](0, 5)[/latex] and [latex](3, 3)[/latex]. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.

Count the rise – it is negative.	The rise is [latex]−2[/latex].
Count the run.	The run is [latex]3[/latex].
Use the slope formula.	[latex]m=\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values of the rise and run.	[latex]m=\frac{-2}{3}[/latex]
Simplify.	[latex]m=-\frac{2}{3}[/latex]
The slope of the line is [latex]-\frac{2}{3}[/latex].

Count the rise.	The rise is [latex]−6[/latex].
Count the run.	The run is [latex]9[/latex].
Use the slope formula.	[latex]m=\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values of the rise and run.	[latex]m=\frac{-6}{9}[/latex]
Simplify the fraction.	[latex]m=-\frac{2}{3}[/latex]
The slope of the line is [latex]-\frac{2}{3}[/latex].

example

Find the slope of the line shown:

Answer: Solution

Locate two points on the graph whose coordinates are integers.	[latex]\left(2,3\right)[/latex] and [latex]\left(7,6\right)[/latex]
Which point is on the left?	[latex]\left(2,3\right)[/latex]
Starting at [latex]\left(2,3\right)[/latex] , sketch a right angle to [latex]\left(7,6\right)[/latex] as shown below.

Count the rise.	The rise is [latex]3[/latex].
Count the run.	The run is [latex]5[/latex].
Use the slope formula.	[latex]m=\frac{\text{rise}}{\text{run}}[/latex]
Substitute the values of the rise and run.	[latex]m=\frac{3}{5}[/latex]
The slope of the line is [latex]\frac{3}{5}[/latex].

Finding the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

• horizontal line [latex]y=b[/latex]; all the [latex]y[/latex] -coordinates are the same.
• vertical line [latex]x=a[/latex]; all the [latex]x[/latex] -coordinates are the same.

Quick Guide to the Slopes of Lines

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