FUNCTIONS AND GRAPHS VII

Characteristics of the afine function

The afine function and the linear function:

Every afíne function:

y = m * x + k

is associated with a linear function:

y = m * x

1.- Modify the parameters m and k and observe the relationship that exists between an afíne function and its corresponding linear function.

How many afine functions are associated with the same linear function?  What can you observe about the straight lines that represent them?

Calculation of the slope of a straight line.

The function:

y = m * x + k

represents a straight line. The parameter m is named slope of the straight line and it indicates its greater or lesser inclination, the same as with the linear function.   

2.- Modify the parameters and move the yellow point.

Notice that the quotient of the differences of co-ordinates between any two points (the length of the green segment divided by that of the blue) is always the slope of the straight line. 

What value would assign to the blue segment so that it is easy to determine the slope?

Representation of the slope of a straight line.

Given the function:

y = m * x + k

if the value of x is increased by one unit, the function is increased in the value of the slope:

3.- Modify the parameters and notice that the length of the yellow segment is the value of the slope. 

The slope is the value that the function increases or decreases when x increases by a unit.  

Check that all the straight lines that are parallel to each other have the same slope.

Representation of the ordinate at the origin of a straight line.

The parameter k is called ordinate at the origin of the afine function because it indicates the value of the function when the value of x is zero.

4.- Check that all the afine functions pass through the point of coordinates (0,k).

Check that the straight lines that pass through the same point of the y axis  have the same value of k and only differ in their slope.

Symmetry with respect to the Y axis.

5.- Find the relationship that should exist between two afine functions so that their straight lines are symmetrical with respect to the y axis.   (Moving the orange point you obtain their symmetries respective to the Y axis.)

Symmetry with respect to the X axis.

6.- Find the relationship that should exist between two afine functions so that their straight lines are symmetrical with respect to the X axis. (Their symmetries with respect to the Y axis are obtained by moving the orange point.)

Author: Juan Madrigal Muga

	Ministerio de Educación, Cultura y Deporte. Año 2000