Mathematics: Exploring the Parabola

When I was introducing algebraic functions to my twelve year old, I summoned the equation of parabola y = ax^2 + bx + c in the hope that it would bring in some visualization to the seemingly drab world of algebra. And so, following the traditional footsteps of teaching, we pegged the values of the constants and derived values of y by substituting values of x and tried plotting them.

It was all going good, till the wonderfully fertile twelve year old brain asked, "What happens to the parabola if a, b and c are not constants?"

Thought it was not very difficult to satiate the twelve year brain, the full fury of the question send me on a mathematical quest late in the night. The realization that in all my years of study, never did I have enough time and patience to 'play' with mathematics was beginning to hurt.

It has since been some months that I have been playing with mathematical modelling tools - especially Maple and Geometry Expressions. These tools have been instrumental in pivoting me beyond the drudgery of derivation to the realm of appreciation of the wonderful world of mathematics.

So, coming back to the question which set all these in motion. The layman answer is that variation of a, b and c moves the parabola (across the cartesian space) and changes its shape (curvature). A more specific answer would be following the loci of the vertex and understanding the changes in curvature.

Variation of 'c'

'c' being an independent variable, the impact on its variation is quite easy to comprehend. It translates the vertex of the parabola on a line parallel to the Y axis. The figure below shows the locus of the vertex as the dotted line.

Variation of 'a'

Variation of 'a' has more drastic effects. For one it moves the vertex along the line y = (b/2)x + c. Also being tied to the square of x, it changes the curvature of the parabola very drastically. I will come to the impact of curvature later in this post. For now, let's focus on the locus of the vertex.

Variation of 'b'

Variation of 'b' is what I would say 'interesting'! While it does change the curvature (though not as drastically as 'a'), it translates the vertex along an inverted parabola y = -x^2a + c. Note that the axis of this inverted parabola will always be the y axis (since b=0).

 

Now, coming to the curvature part - this needs some higher order mathematics. 

Radius of curvature of a curve y = f(x) is 

For the parabola the radius of curvature becomes

Differentiating the radius of curvature w.r.t. a and b separately, we get:

 

To see the effect of variation of a and b, we peg the values of x at 1. The graphs below show the effect. Clearly, the impact of varying a is lot greater than b as expected.