This question is intended as an extension of a previous question:

http://www.experts-exchange.com/Miscellaneous/Math_Science/Q_21081042.html

And, because of generous contributions made by PointyEars in the above link, I would like to designate 100 points to PointyEars (provided some comment is made here).

I am looking for general information regarding standard forms,

For example:

How many "standard forms" are there?

For the parabola:

(x-h)² = 4p(y-k) - http://www.gomath.com/Questions/question.php?question=13281

y=a(x-p)²+q - From my text.

y = A*x² - From snoyes_jw

y = A*x² + B

I want to be able to recognize variations of standard forms. I also want to make it interesting. The comments made by PointyEars in the link above did just that. Hoping for more of the same.

Thanks

http://www.experts-exchange.com/Miscellaneous/Math_Science/Q_21081042.html

And, because of generous contributions made by PointyEars in the above link, I would like to designate 100 points to PointyEars (provided some comment is made here).

I am looking for general information regarding standard forms,

For example:

How many "standard forms" are there?

For the parabola:

(x-h)² = 4p(y-k) - http://www.gomath.com/Questions/question.php?question=13281

y=a(x-p)²+q - From my text.

y = A*x² - From snoyes_jw

y = A*x² + B

I want to be able to recognize variations of standard forms. I also want to make it interesting. The comments made by PointyEars in the link above did just that. Hoping for more of the same.

Thanks

A*x² + B*y² + C*x*y + D*x + E*y + F= 0

if we evaluate

| A C/2 D/2|

|C/2 B E/2| =H

|D/2 E/2 F |

| A C/2|

|C/2 B | = J

then

when H != 0 and J < 0 we have a hyperbola

when H != 0 and J = 0 we have a parabola

when H != 0 and J > 0 we have an ellipse

When H = 0 and J < 0 we have intersecting lines

when H = 0 and J > 0 we have a point

when H = 0 and J = 0 we have parallel lines

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This comment is a detailed analysis of some (not all) quadratic equations. I cannot invest more time in it, but I hope that you will find it helpful...

All the curves we have been talking about are polynomial. That is, of the type:

sum( A_kn * x^k * y^(n-k) ) = 0 sum for all k = 0 .. n and for all n = 0 .. N

where A_kn are numeric coefficients (i.e. constants) dependent on k an n, and N is an integer called the degree of the equation. This is the most generic form of such equation you can write.

Each term has a degree given by the product of the exponents of x and y. Obviously, some of the coefficients can be zero (i.e. some of the terms can be missing), but if all the terms of degree N are missing, the equation "degenerates" into an equation of degree N-1.

The equations of degree 0 are no equations at all. They degenerate into the identity 0 = 0.

All possible equations of degree 1 are:

a*x = 0

a*x + c = 0

b*y = 0

b*y + c = 0

a*x + b*y = 0

a*x + b*y + c = 0

When plotted on the xy axes, these equations are all straight lines. The most generic is the last one. The first one is the equation of the y-axis, the second one of a parallel to the y-axis, ..., the second last is a line passing through the origin at an angle.

We have been talking of equation with N = 2 (i.e. quadratic). The full list of possibilities for N=2 (not degenerate) consists of equations that we can group as follows:

=== only x

a*x² = 0

a*x² + b*x = 0

a*x² + c = 0

a*x² + b*x + c = 0

These equations are solvable in x and can be written as (x - x1)*(x - x2) = 0, with x1 possibly = x2. This means that, when plotted, these equations produce two (or just one) straight lines parallel to the y-axis: x = x1 and x = x2.

=== only y:

Like case 1 but with line[s] parallel to the x-axis

=== both x and y:

The most generic equation in this group is:

a*x² + b*y² + c*x*y + d*x + e*y + f = 0

There are 6 terms in total. All possible equations with 2 terms :

a*x² + b*y² = 0 => sqrt(a)*x = ± sqrt(-b)*y => two lines passing through the origin

a*x² + c*x*y = 0 => x * (a*x + c*y) = 0 => y-axis and one line through the origin

a*x² + e*y = 0 => parabola with axis on the y-axis and apex in the origin

b*y² + c*x*y = 0 => y * (c*x + b*y) = 0 => x-axis and one line through the origin

b*y² + d*x = 0 => parabola with axis on the x-axis and apex in the origin

All possibile equations with 3 terms. These are quite a few...

a*x² + b*y² + c*x*y = 0

can be written as

y² * [a*(x/y)² + b + c*x/y] = 0

The expression in square brackets can be solved in x/y:

y²*(x/y - z1)*(x/y - z2) = 0

with z1,z2 constants =>

(x - z1*y)*(x - z2*y) = 0 => two straight lines

a*x² + b*y² + d*x = 0

if you replace x with t - d/(2a) you get

a*t² + b*y² = d²/(4a) => ellipse centred in (d/2a, 0) (circle if a = b)

a*x² + b*y² + e*y = 0 => ellipse centred in (0, e/2b) (circle if a = b)

a*x² + b*y² + f = 0 => ellipse (circle if a = b)

a*x² + c*x*y + d*x = 0

can be written as

x * (a*x + c*y + d) = 0 => two lines

a*x² + c*x*y + e*y = 0 => ZZZ

a*x² + c*x*y + f = 0 => ZZZ

a*x² + e*y + d*x = 0 => a parabola with axis parallelto the y-axis

a*x² + e*y + f = 0 => ditto

b*y² + c*x*y + d*x = 0 => ZZZ

b*y² + c*x*y + e*y = 0 => two lines

b*y² + c*x*y + f = 0 => ZZZ

b*y² + d*x + e*y = 0 => a parabola with axis parallelto the x-axis

b*y² + d*x + f = 0 => ditto

For the equations marked with ZZZ I don't immediately identify the curve. Perhaps I will work on them later (but don't hold your breath! :-)

With 4 terms, the analysis becomes more difficult. All possible equations with 4 terms containing both x and y are:

a*x² + b*y² + c*x*y + d*x = 0

a*x² + b*y² + c*x*y + e*y = 0

a*x² + b*y² + c*x*y + f = 0

a*x² + b*y² + d*x + e*y = 0

a*x² + b*y² + d*x + f = 0

a*x² + b*y² + e*y + f = 0

a*x² + c*x*y + d*x + e*y = 0

a*x² + c*x*y + d*x + f = 0

a*x² + c*x*y + e*y + f = 0

a*x² + e*y + d*x + f = 0

b*y² + c*x*y + d*x + e*y = 0

b*y² + c*x*y + d*x + f = 0

b*y² + c*x*y + e*y + f = 0

b*y² + d*x + e*y + f = 0

There are only 6 choices of 5-term equations, each obtained by leaving out one of the 6 terms.