Linear Functions

Hello, class. I am Professor Carlos Torres, but you can just call me Carlos. Today we will be discussing Linear Functions. After we cover this lecture I hope that you will be able to understand what linear functions are clearly and dominate the material for the next test. Now, lets get started. First of all, a linear function is basically an equation that shows that the relationship between the dependent variable (y) and the independent variable (x) of a function.  It is represented by the formula f(x) = ax+b. Now, not not all relationships between the dependent and independent variables represent linear functions. For a function to be considered linear, both the x and y values must have a constant rate of change. If the function does not have a constant rate of change in both variables then it is a different kind of function. For it to be considered linear, the constant rate of change between the variables is necessary. Down below are a couple of examples of what linear functions would look like:

Do you see how in each of the x and y values in all of the tables have constant rates of change? In the first table of values, the values of x are constantly increasing by a number of two while the values of y are constantly increasing by a value of two as well. This is how you can tell if a function is linear or not. Next, I will demonstrate to the class an example of what a linear function isn't. 

On this graph, the values of x do have a constant rate of change but the y values do not. First a seven is added, then a six, then an 8 and then another six. This cannot be considered a linear function. 

Now, I would like to show the class how a linear function would be seen throughout a graph. 

Do you see how the table of values, when drawn, represents a straight line? This is due to the fact that both the x and y variables have respective constant rates of change of 2. 

Now lets do a problem to see if you understood the material. Determine which of the table of values is a linear function and which one is not? 

Based on what we know, on the first graph we can see that the x values are constantly decreasing by a value of 1 while the y values are constantly increasing by a value of 1. The variables clearly have a constant rate of change, thus this is a linear function.

On the other hand, on the second table of values we can see how the x values go from 0-2, 2-4, 4-8 and 8-10, as you can see, the values do not have a constant rate of change because from 4-8 there is an added value of 4. The y values do not have a constant rate of change either, this is why the second table of values is not a linear function. 

Linear equations are relatively simple to understand by just paying attention to the change in values throughout the table of values, does anybody have any questions?