Slope and Rate of Change
 Topic Index | Algebra Index | Regents Exam Prep Center

Slope and Rate of Change

The word slope (gradient, incline, pitch) is used to describe the measurement of the steepness of a straight line.  The higher the slope, the steeper the line.  The slope of a line is a rate of change.
                     

The building code for using asphalt shingles on roofs states that the minimum pitch must be a rise of 4" for every 12" of horizontal distance (run) covered.  Asphalt shingles are not to be used on roofs that have very little steepness.  Builders check to see if the pitch (slope) of the roof is before using asphalt shingles.

 

Builders need to know the pitch of a roof to determine which type of shingle will be appropriate for the roof.
Slope is a ratio and can be expressed as:

change in y
 over
change in x.

 or 

or 

or

 

Let's examine the slope of a straight line more carefully.

Consider the line  y = 2x + 1, shown at the right.

By how much has the value of y changed between the two points (-4,-7) and (-3,-5)?  This will be a vertical change.       Answer:  2 units
 (as read from the left point to the right point, with the right point being "higher" on the graph)

By how much has the value of x changed between the two points (-4,-7) and (-3,-5)?   This will be a horizontal change.   Answer:  1 unit
(as read from the left point to the right point on the graph)

The slope = = = 2
 

Notice that this slope will be the same if the points (1,3) and (2, 5) are used for the calculations.  For straight lines, the rate of change (slope) is constant (always the same). 


For every one unit that is moved on the x-axis, two units are moved on the y-axis.  This is true at any location on the line.


 

Example:

Find the slope of a line passing through the points (-4,4) and (8,-2).

You have several choices:
1.  You can graph the points and "count" the vertical changes and horizontal changes to use in the formula:
        
            Slope =
 


Notice that to read the rise and run for these two points, we started at (-4,4), moved "down" (negative) 6 units and moved "right" (positive) 12 units.
2.  You can substitute the points directly into the formula
                 m =    =   

 

 

Example:
    Todd had 5 gallons of gasoline in his motorbike.  After driving 100 miles, he had 3 gallons left.  The graph at the right shows Todd's situation.

a.  Find the slope of the line.
      

b.  What does this slope tell us?
Since , we know that Todd's bike is burning .02 gallons of gasoline for every mile that he travels.   The negative value of the slope tells us that the amount of gasoline in the tank is decreasing.

c.  What is Todd's mpg?
The    tells us that Todd can drive 50 miles on one gallon of gasoline (an mpg of 50 miles per gallon). 

While slopes of lines are not labeled with units, rates of change used in application problems often take on the units depicted in the problem, such as Todd's bike burning .02 gallons per mile,
or his mpg being 50 miles per gallon.