In my earlier post on Linear Function, I described how a linear function can be recognized based on equation, graphs, and tables. In this article, let us talk about how to derive the equation of linear function (as they are called in calculus) or the equation of a line (as they called in analytic geometry). I will be using linear function and line interchangeably in the discussion as most teachers would do. It is important to note and to make sure that students are aware that when the topic is on linear function and the teacher says “line” as in “What is the equation of this line?” it actually mean, “What is the equation of the linear function represented by this line?”
What is the equation of linear function?
Textbooks define a linear function as a function defined by the equation y=ax+b (or y=mx+b as m is commonly used for this form) or the equation ax+by+c=0. The latter equation is called the standard form of the equation of a line. In my opinion, this form should not be used when talking about equation of linear function because it does not show clearly the relationship between x and y. The equation y=ax+b shows the relationships between the independent variable x and the dependent variable y, where the value of the variable y is determined by the rule ax+b. But as I said earlier, in school mathematics, this is used interchangeably since we can transform one to the other by algebraic manipulation and we would be getting the same set of points from both. For example the equation 2x-y-3=0 could be transformed to y=2x-3.
Where did y=ax+b come from?
In my previous post about linear function I introduced linear function as a polynomial function of at most degree 1 so ax+b defines the linear function x→ax+b (also expressed as y=ax+b), where a and b are constant. In this case, linear function is defined based on the structure of the equation defining it. Do you think this would make sense to students? Telling is never a good way of teaching mathematics. I propose here a simple activity that would lead to the derivation of of the linear equation of a linear function from graphs.
The Cartesian plane is made up of points and each point is named by the ordered pair (x,y). In the figure on the right, there is something special about the points E, D, C, F, B, G, and A. They all belong to one line. They are collinear. What does it take for a point to be a member of this elite group? Can you just think of any point and say that it belongs to that group? Observe the x and y coordinates of the points on the line. What do you think would the coordinates of the middle point between D and C? How did you get that? What about the middle point between E and D? What would be their coordinates? If you think you have discovered the condition for membership on this line, try more points. Find the coordinates of the midpoints between C and F, F and B, B and G and G and A.
Challenge: Does the point P=(-12,-20) belong to the line where points A, B, C, D, E, F and G are?
Question to the teacher-reader: How would you proceed from here to derive the equation y=mx+b?
Next: How to derive the equation of linear function from its graph (This is the continuation of the Activity presented in this post.)