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Rotating Parabolas

Date: 01/19/99 at 21:36:59
From: Molly Wilson
Subject: Parabolas that are not functions or inverses of functions

I have come to notice that all parabolas that I come across in my 
algebra II class have a line of symmetry that is either a horizontal 
or a vertical line. The simplest equations or parabolas are y=x^2 and 
x=y^2. I was curious to find out how one would come up with the 
equation or a parabola with a line of symmetry such as y=x or some line 
that is not parallel to the x or y axis. I asked my algebra II 
teacher, who said it was possible, but didn't know how to come up with 
an equation. The ways I tried to adjust the parabola equation all 
seemed to result in the same sort of parabola with a horizontal line 
of symmetry or in a split parabola, or in a different sort of curved 
line. Any sort of information you could provide in leading me to a 
solution would be greatly appreciated. Thank you.

Date: 01/20/99 at 17:06:10
From: Doctor Peterson
Subject: Re: Parabolas that are not functions or inverses of functions

Hi, Molly. This is a great question! You're really thinking.

As you pointed out, the equation you're looking for will not be a 
function, which is probably why you aren't taught about it at your 
level. What you need to do is take the familiar equation y = Kx^2 and 
rotate it about the origin.

              y         Y
               \        |
                \       |    *
                 \      |      (x,y)
                  \     |     *
                   \    | /    \
                    \ / |     * \       / x
                     \  |        \  /
       *              \ |   *   /
          *            \|   /angle t
                   */   |\
                /       | \
            /           |  \
        /               |   \
                        |    \
                        |     \
                        |      \
                        |       \
                        |        \

The way to do this is to make a substitution of two new variables 
(I'll use X and Y) using equations like this:

    x = cos(t)*X + sin(t)*Y      
    y = -sin(t)*X + cos(t)*Y

where sin and cos are the sine and cosine functions from trigonometry, 
and t is the angle by which you area rotating the parabola. In case 
you don't know trig yet (and to simplify my work), I'll avoid that by 
letting you just choose any two numbers A and B and define

    x = AX + BY
    y = -BX + AY

This will both rotate and enlarge or reduce the graph, but it will 
still be a parabola. So let's do it:

    y = Kx^2


    -BX + AY = K(AX + BY)^2

which we can simplify to

    -B X + A Y = KA^2 X^2 + 2ABK XY + KB^2 Y^2

For example, if I set A, B, and K to 1 (using a 45 degree rotation), I 

    -X + Y = X^2 + 2XY + Y^2

You might like to solve this for Y in terms of X (using the quadratic 
formula). You'll find you have (in general) two values of Y for each 
X, and two X's for each Y.

In general, a conic section will have an equation like this:

    A x^2 + B xy + C y^2 + D x + E y + F = 0

and with certain conditions on the constants, it will be a parabola.

I'll leave it at that and let you play with it - there's a lot here to 
get your mind into!

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Equations, Graphs, Translations

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