A function where the response variable is equal to a constant times the explanatory variable is called a directly proportional function. Symbolically, with y as the response and x as the explanatory variable, with a constant c, we have:
y = c * x
Do you recall some other "famous" functions which are directly proportional, like:
Directly proportional functions have graphs which are straight lines. In addition, since the response variable is equal to 0 when the explanatory variable is 0, the graph of a directly proportional function passes through the origin, with coordinates (0 , 0).
More generally, a linear function is any function whose graph is a line, and not just a line passing through the origin. Symbolically, with y as the response and x as the explanatory variable, and with a constants c and d, we have:
y = c * x + d
In the linear function above, the constant c represents the rate of change. It is also called the slope of the line. When data have a constant rate of change, no matter which two points you choose, the change in the response (dependent) variable divided by the change in the explanatory (independent) variable will be constant.
So, if we pick two points 
and 
and calculate the rate
of change, it should be a constant:
Previously, we considered situations where the response (or dependent) variable was determined completely by just one other variable. Now we will consider cases where the one response variable depends on several explanatory (or independent) variables. In addition, we will examine the nature of this dependence.
Weight up to
|
Rate (in dollars) |
| 1.0 | 0.60 |
| 2.0 | 0.85 |
| 3.0 | 1.10 |
| 4.0 | 1.35 |
| 5.0 | |
| 6.0 | |
| 7.0 | |
| 8.0 |
(a) If we only consider integer numbers of ounces, verify that the table represents a linear function.
(b) Fill in the remaining table entries.
(c) How would you represent this function graphically?
In our next lecture on Topic 5, we will discuss how rates of change are related to the shape of a function. The sign of the rate of change (or slope) indicates whether the function is increasing or decreasing, and the trend of the rate of change (whether it is increasing or decreasing) indicates whether the function is concave-up or concave-down.