Unit D: Graphing

Rate of Change and Slope

Is it possible to find the rate of change when given points on a graph. The next example will demonstrate how rate of change and slope are related.
 
An administrator of a school is monitoring the weather to determine if the school should be closed for the day. At 9:00 p.m., there is no snow on the ground but it has started to snow. It is predicted that 2 cm of snow will fall each hour until 7:00 a.m.

  1. What is the expected rate of change in the depth of snow?

  2. Complete the table of values for the predicted amount of snowfall starting at 9:00 p.m., which represents 0 h, and ending at 7:00 a.m., which represents 10 h. Does the table of values demonstrate direct variation? Explain.

    x y
    0 0
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10


  3. Identify the dependent variable and the independent variable.

  4. Graph the table of values. Should the points be connected? Explain.



  1. The forecast is for snow to fall at a rate of 2 cm per hour. The rate of change is 2 cm/h.






  2. Yes, the table of values represents a direct variation relationship. When the x-value increases by 1, the y-value increases by 2.

    Each value of x is multiplied by a factor of 2 to produce each y-value.

  3. The snow accumulation (height of snow) is dependent on the time. Therefore, time is the independent variable and snowfall (height of snow) is the dependent variable.

  4. There will be snow accumulation between the hours represented on the graph. For example, at 1.5 hours, the height of the snow increases. Therefore, the set of data is continuous and the points are connected. 




On the Snow Accumulations graph, it is evident that when the x-values increase by 1, the y-values increases by 2. This demonstrates direct variation.

The slope, or rate of change, is the steepness of a line on a graph. The slope is calculated using the formula

m = rise run = change   in   y change   in   x

The letter "m" is used to represent slope. The formula most commonly used to find slope is m = rise run .




The slope, m, of the line on the Snow Accumulation graph is

m = rise run = 2 1 = 2
In this example,
  the   rate   of   change = slope = 2

The slope can also be calculated if the coordinates of any two points (x1, y1) and (x2, y2) on a graph are known.

m = y 2 - y 1 x 2 - x 1

where 
  • y2 is the y-coordinate of point 2.
  • y1 is the y-coordinate of point 1.
  • x2 is the x-coordinate of point 2.
  • x1 is the x-coordinate of point 1.




Calculate the slope given two points on the graph: A(3, 6) and B(5, 10).




Let A(3, 6) = (x1, y1) and B(5, 10) = (x2, y2).

m = y 2 - y 1 x 2 - x 1 = 10 - 6 5 - 3 = 4 2 = 2

The slope, m, is the rate of change, which is 2.
 
Skill Builder

Skill Builder

For more information on finding slope from a graph, click the Skill Builder button to access the Skill Builder page.