Slope of a Line

In Graphing Proportional Relationships we see how the slope of the line generated when a linear relationship is plotted represents the unit rate e.g. miles/ hour, cost/ mile, etc. The slope of a line can be represented using a positive or negative number to show its steepness and direction. The steepness is sometimes referred to as the rate of change.

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Slope Examples

The table and graph below shows x and y values resulting from the function y = 2x.

y = 2x
x
y
-5 -10
-4 -8
-3 -6
-2 -4
-1 -2
0 0
1 2
2 4
3 6
4 8
5 10

graph showing algebraic relationship y = 2x.

The slope of the line above is 2. Each value of y is 2 times the corresponding value of x.

Look at the equations and resulting tables and graphs below.

y = 3x
x
y
-3 -9
-2 -6
-1 -3
0 0
1 3
2 6
3 9

graph showing algebraic relationship y = 3x.

y = x/2
x
y
-10 -5
-8 -4
-6 -3
-4 -2
-2 -1
0 0
2 1
4 2
6 3
8 4
10 5

graph showing algebraic relationship y = x/2.

Notice also that in the above examples the line slopes up from left to right. These are positive slopes show that as x increases, so does y.

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Negative Slope

The examples below show that as x increases, y decreases. This results in a negative slope that runs downwards from left to right.

y = -x
x
y
-10 10
-8 8
-6 6
-4 4
-2 2
0 0
2 -2
4 -4
6 -6
8 -8
10 -10

graph showing algebraic relationship y = -x.

y = -2x
x
y
-5 10
-4 8
-3 6
-2 4
-1 2
0 0
1 -2
2 -4
3 -6
4 -8
5 -10

graph showing algebraic relationship y = -2x.

Graph Generator!

Generate a variety of lines using the y = mx + b format for linear equations using the boxes below to enter values for m and for b.

Your browser does not support the HTML 5 Canvas.

Enter slope(m) and y-intercept(b) below then click Draw Line


Click Draw Line to graph the equation

How To Calculate Slope

We have seen already that if a linear relationship is given as an equation in the form, y = mx + b, then the constant m represents the slope. Often though a linear relationship is given as sets of related values that can be graphed as ordered pairs. See the example below:

x
y
-5 -7
-4 -5
-3 -3
-2 -1
-1 1
0 3
1 5
2 7
3 9
4 11

graph showing algebraic relationship y = 2x + 3.

If you know any two points on a line you can calculate the slope:

Slope = difference in y-values/ difference in x-values (rise/ run)

Slope = (y2 - y1)/ (x2 - x1)

annotated line with 2 points, (x1,y1) and (x2,y2) shown with difference in x and in y values shown.

Using points (-3,-3) and (1,5) from the table above we get:

Slope = (5 - (-3))/ (1 - (-3))

Slope = 8/4

Slope = 2

You will find more examples showing the use of the slope formula here.

Any 2 Points Will Do - Similar Triangles

As the example above shows, slope can be calculated using the co-ordinate values of any two points on the line. Note below how any two pairs of points (that’s two sets of two!) on the same line form similar triangles on the coordinate grid. The angles forming the slope in similar triangles are the same in each triangle. There is more on similar triangles and their geometric properties here.

line on Cartesian grid with various points marked on line together with the triangles they form on the grid.

Look at points A and B. They form triangle BAE on the grid. Points C and D form triangle DCG. △BAE and △DCG are similar triangles as their three internal angles match either other. Each of these two triangles are similar to △DBF. In fact any right triangle with its hypotenuse lying on the slope line would be similar.

Worksheets

Use the worksheet(s) below for practice.

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