Function – Straight Line

 

    Motivation:        y=f(x)=mx+c , x  R

                        For each x, there exists corresponding y .   Collect the data and draw it.

                                                   

 

                         It looks like a straight line. However,  are all the straight lines  with the            form y=mx+c ?

 

                       We just use a concept from Geometry:

                                 

                     If  L1, L2, and L3 are parallel to each other. The line segments they intercept have the following properties:

 

                                          =

 

 

                     Thus,  use the following plot:

                                          

                                 If a line passes the points  (a,b) and (c,d ),  then we have the following relationship:

 

                                               ( for b-d  0 )

                                     or

                                               ( for a-c  0 )

 

                      Thus, we are sure that any straight line equation has those forms.

                      For the lines that are not perpendicular to x-axis, they can be simplified as

                                            

                                                                    y=mx+t

 

                             

Theorem ( Line Equation )

                       Given two points (a,b) and (c,d),  the line equation to pass the two points is

                                          ( for a-c  0 )