FUNCTIONS AND GRAPHS III

The linear function

The identity function.

y = x

is named the identity function because each number on the X axis corresponds to the same number on the Y axis.

1.- Move the orange point and see how all the points which fulfill this condition y = x are on the straight line. Change the scale and verify it for large and small values of x.

The graphical representation of the identity function is a straight line.  It is the bisector of the first and third quadrants of the Cartesian system of reference.

The double function:

y = 2x

2.- Observe the points of the double function. (You can move the point with the red and blue arrows or also directly dragging the orange point or the blue with the mouse.   Also you can move the segments dragging the extreme left)

What will the graph of the double function be like? Draw it in your work book.

Comparison of functions:

3.- Observe the graphs of the identity function and the double function.

What will the graphs of the triple, quintuple, half, sixth part, tenth part, etc.functions be like?

Linear functions.

Functions which are of this form, that is to say that the value of  y is equal to a real number multiplied by the value of x, are called linear funtions. 

y = m*x

4.- Represent various linear functions: y = 3x; y = 5x; y = ½ x, y=-2x, y=-3/4x; etc. (Move the red point and see how the straight line and its equation change.)

Points of linear functions.

The straight line which represents a linear function is made up of infinitely many points, and all of them fulfil the condition:

y = m*x

5.- Choose a straight line and see how, on moving the red point over it, the corrdinates of the points maintain the relationship  ordinate = m * abscisa. (The last decimal figure is rounded up so that two decimals are always shown, and so this figure will notalways  agree exactly.)

Analysis of the linear functions.

6.- Change the value of m and observe what is common to all linear functions and what differences they may have. Write your conclusions in your work book.

What name do you think is appropriate to designate what the letter m represents?

- ascent (increase) of the sraight line

- descent (decrease) of the straight line

- slope of the straight line

- inclination of the straight line

- horizontalness of the straight line

- verticality of the straight line

- ?

The slope of the straight line.

7.- The value m is called the slope of the straight line.

Analyse what happens for big values of the slope, for values near zero and for negative values.  Write your conclusions in you work book. 

Could the straight line represented by a linear function be vertical? Why?  What value should the slope have?

Author: Juan Madrigal Muga

	Ministerio de Educación, Cultura y Deporte. Año 2000