FUNCTIONS AND GRAPHS VII

The quadratic function

The family of parabolas y=k*x^2

The function y=k*x^2, with k being a real number, is a family of parabolas. For k=1 it coincides with the function y=x^2.

Press the Init button and afterwards write the following values of k: -1, 2, -2, 3, -3, 0.01, -0.01, 50, -50. Is the graph of the function for k=0 a parabola? When are the branches of the parabola more open, when the absolute value of k is greater or less than 1?  If k is positive, towards where do the branches of the parabola point? And if k<0?

The function y=k*x^2+b

If we represent the previous quadratic function for different values of the real number b, we will see that it is a translation of the parabola y=x^2 along the Y axis for which the vertex is found at the point (0,b).

Represent the function y=3*x^2+b for the following values of b: 1, -1, 2, -2. Where does each one have its vertex? Do any of them intersect the X axis? Where?
Use the Clear button and represent the functions y=k*x^2-3 clicking the arrows above and below k. Where do all these graphs have their vertex? Where do they cut the X axis?

The function y-b=k*(x-a)^2

We are going to check that a quadratic function of this type is a parabola with the vertex at the point (a,b).

Represent the following second degree functions: y=2*(x+3)^2-2, y=-(x-1)^2+1, y=x^2, y=-x^2, y=0.75*(x+0.5)+0.25. Where does each one of them have its vertex?
Represent the function y=x^2+2*x+1 bearing in mind that (x+1)^2=x^2+2*x+1 Where does it have its vertex?

The second degree function y=a*x^2+b*x+c

In general, any function of this form is a parabola.  We are going to use the Descartes program in a way that the function that we want to represent can be written directly. 

Represent the following second degree functions and try to investigate the coordinates of the vertex and the points of intersection with the axes: y=x^2+3, y=x^2-2*x, y=x^2+2*x+3, y=-x^2-4*x-2.

Author: Miguel García Reyes

	Ministerio de Educación, Cultura y Deporte. Año 2000