FUNCTIONS AND GRAPHS VI

The most simple square function

The function y=x^2

The graph of the function y=x^2 is what is called a parabola, a curve that has an important physical meaning  because it describes the course of a projectile, like a bullet or ball without air friction.  The parabolic mirrors reflect the luminous rays from a light source situated in the focal point as rays parallel to each other.  These spotlights are used in car headlights and as aerials in radioastronomy, and satellite television.       

Move the point P with the mouse to the right and left and watch its coordinates.  Make a table of values and try to represent the graph of the function.     

The graph of y=x^2

On representing the table of values of the function on some cartesian axes we will obtain an approximation of the graph of the most simple square function.  

Modify the values of x for values greater and less than zero, clicking on the arrows.  Compare the form of the graph with that which you have drawn previously.  Decrease the scale to 5 and modify the values of x again. Press the Init button and increase the scale to 500. Change the values of the abscissa again from negative to positive.  Make them represent intermediate points between those already represented, for example, 0.05, 0.15, -0.05, -0.15, etc.

The parabola

Joining the points represented previously the graph of the function y=x^2 is obtained, it is called a parabola.

Change the values of x with the arrows and watch the movement of the point that belongs to the curve and the different values of the abscissa and ordinate.   Modify the scale to 5 and watch the graph.  

Symmetry

For negative values of x the value of the function is the same as for the corresponding negative values.  This means that the curve is symmetrical with respect to the ordinate axis.  The Y axis is, in this case, the axis of symmetry of the parabola, that is to say the place by where we would be able to fold a parabola drawn on paper so that its two halfs, called branches of the parabola, coincide.  The place where the parabola intersects the axis of symmetry is called the vertex.  The vertex of the parabola y=x^2 is the origin of coordinates.            

Check that for negative values of x the function reaches the same value. Write in your notebook the coordinates of the vertex of the parabola y=x^2 and the equation of the axis of symmetry. 

Growth and decrease of the function.

A function f(x) is said that it is increasing at an interval [a,b] if any two values x1, x2 of the interval such that x1<x2, fulfil that f(x1)<f(x2). If it is fulfilled that f(x1)>f(x2) then it is said to be decreasing. If, finally, f(x1)=f(x2) it is called constant.

Study whether the function y=x^2 is increasing, decreasing or constant at the intervals: [0,1], [-10] and [-1,1] What happens for values of x greater than zero? And with the negative values?

Rate of average variation of a function

The Rate of Average Variation of a function f(x) between two points x1 and x2 is the quotient:

TVM=((f(x2)-f(x1))/(x2-x1)

A function f(x) is increasing, decreasing or constant at a determined interval [a,b] if for any two values of x which are inside the interval, the Rate of Average Variation of the function is positive, negative or zero. 

Analyse the growth or decrease of the function being studied observing the TVM at the following intervals: [0.5,1], [-0.5,0],[10,11] and [-11,-10]. In the last two cases change the scale to 5 moving conveniently the axes.   

The minimum

When a function passes, at a point, from being increasing to decreasing it is said that it has a minimum at this point. When it passes from being decreasing to increasing at this point, it has a maximum in it.

The function y=x^2 has a maximum or minimum. At what point?

Author: Miguel García Reyes

	Ministerio de Educación, Cultura y Deporte. Año 2000