Department of Math Education
J. Wilson, EMAT 6680

The Parabola

Lesson #2 - by Jan White

What happens to y=ax² + x +2 as a is varied?

As a changes the graphs can be seen to shrink or expand.  When the absolute value of a is greater than 0 then the graph shrinks towards the y-axis and when the absolute value of a is less than 0 then the graph expands towards the x-axis. When a is negative, this causes the graph to open downwards.  The dialation would remain the same.

What happens to y=x² + x +c as c is varied?

As can be seen by the graphs above, when c is varied this will raise or lower the graph determining the y intercept.

What happens to y=x² + bx +1 as b is varied?

As b is varied you can see that these graph are both dialated and transposed up and down.  If we look a little closer there seems to be a correlation among the vertexes of theses graphs.
The yellow graph, -x² +1, above is the locus of the previous graphs as b is varied.

Summary:

As a is varied and b and c are fixed the parabolas become dialated.  When c is varied and a and b are fixed this either raises or lowers the parabolas. As b is varied this will decribe the locus of the parabola y = -x² + c. This is something that would be time consuming in the classroom to do by hand but is very effective when done by this graphing program.