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Rate of Change & Slope of a Line

The examples below show how the rate of change in a linear function is represented by the slope of its graph. The formula for calculating slope is explained and illustrated.

If required, you may wish to review this Coordinate Graphing Lesson before working through the examples below that show how the slope of a line represents rate of change.

Formula for the Slope of a Line

The slope of a line is calculated by dividing the rise by the run. The "rise" and the "run" can be found using any two points on the line as shown in the examples below.

Note: The letter "m" is used to represent slope in equations.

 

 

Calculating the Slope of a Line

Slope (m) = Rise
Run
illustration of slope being rise divided by run
l

The rise equals the difference in the y-coordinates.

The run equals the difference in the z-coordinates.

So the formula for slope of a line is:

m = Y2 - Y1
X2 - X1
slope on a coordinate grid shown as being y2 - y1 divided by x2 - x1

Example

Calculate the slope of the line using the two sets of coordinates.

coordinates 2,5 and -2,-3 joined by a line on a coordinate grid

Substitute the values into the formula and solve.

m = 5 - (-3)
2 - (-2)
  = 8
4

 

m

 

=

 

2

Note: It does not matter which pair of coordinates you call (X1, Y1) and which pair you call (X2, Y2) as long as you use the same order when subtracting to find the rise as you do when subtracting to find the run.

For the equation of a line, y = mx, the value m represents the slope. The slope (m) of the line that is generated can be given as a positive or negative number which shows its steepness and direction. The four examples below show the slope for different linear functions. These functions are shown in algebraic, tabular, and graphical form.

Slope of a Line : Example 1

The slope of the line below is 3. The value of y increases 3 times as much as does the value of x.

y = 3x

x y
0 0
1 3
2 6
3 9
4 12
xy graph with y = 3x slope plotted

 

 

Slope of a Line : Example 2

The slope of the line is this example is 1/2 or 0.5. y increases by 0.5 for every increase of 1 in x.

y = x/2

x y
0 0
2 1
4 2
6 3
8 4
xy graph with y = x/2  slope plotted

In both Example 1 and Example 2 above, the line slopes upward from left to right. These are positive slopes or positive rates of change. As x increases, y also increases. Note the difference in Examples 3 and 4 below. They show that, as x increases, y decreases. This results in a negative slope that runs downwards from left to right

Slope of a Line : Example 3

The slope of the line below is -1.

y = -x

x y
0 0
2 -2
4 -4
6 -6
8 -8
xy graph with y = -x slope plotted

 

Slope of a Line : Example 4

The slope of the line in this example is -2

y = -2x

x y
-4 8
-2 4
0 0
2 -4
4 -8
xy graph with y = -2x slope plotted

 

Rate of Change

In the examples above the slope of line corresponds to the rate of change. e.g. in an x-y graph, a slope of 2 means that y increases by 2 for every increase of 1 in x. The examples below show how the slope shows the rate of change using real-life examples in place of just numbers.

Rate of Change: Example 1

The graph below shows the speed of a vehicle plotted against time. Look at the different slopes - both the steepness and direction. These represent the rate of change of speed otherwise known as acceleration.

What is happening?

A - B: Starts from stationery position and increases speed up to 50 m.p.h after 10 seconds.

B - C: Traveling at a constant 50 m.p.h. for 15 seconds.

C - D: Slowing down (probably breaking) to 20 m.p.h. over 5 seconds.

D - E: Speeding up from 20 to 30 m.p.h. over 15 seconds.

E - F: Slowing down and stopping over 5 seconds.

rgraph showing speed against time with various rates of change

 

Rate of Change: Example 2

Michelle has a bank account she uses for just one thing - paying $50 every month to her favorite charity. The bank account has $800 in it at the beginning of the year.

The slope of the graph below shows the rate of change in the bank balance. The slope is -50 which corresponds to the $50 per month that is coming out of the account.

graph showing $50 decline per month in bank balance

 

Final Note: Watch the Scales of the X and Y Axes.

Look at the slope in the example below and compare it to Example 2 above. Which slope is steepest? Which shows the greatest rate of change?

bank balance against months showing a decline with different y-axis scale

Both graphs show a decline of $50 per month. They both show the same rate of change. It is only the difference in scale of the y-axis that makes Example 2 appear steeper.

Sometimes people wish to emphasize or de-emphasize rates of change (e.g. employment rates, change of price) and they can try to do so by choosing whatever scale they like for the axes of the graphs.

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