General form of the quadratic equation

[jimalex]

I am trying to figure out how the coeffecients in the general second degree equation transform the function as they change through a range of values. I think that d and e are the proportion of x and y are horizontal and vertical shifts, but less sure about a, b, and c. Here is the equation as i am familiar with it

ax^2 + bxy + cy^2 + dx + ey + f = 1

I'm having trouble finding good material online, maybe just don't know where to look.

Also, i would like to find some free software similar to geometers sketchbook, particularly where i could create interactive graphs. I've seen some people have made that have sliders that allow you to change a value on the graph as see the change it causes. Anybody know of any?

Thanks, all.

 

[haruspex]

I am trying to figure out how the coefficients in the general second degree equation transform the function as they change through a range of values. I think that d and e are the proportion of x and y are horizontal and vertical shifts, but less sure about a, b, and c. Here is the equation as i am familiar with it

ax^2 + bxy + cy^2 + dx + ey + f = 1

I think you mean = 0.
The different coefficients do not have easily separable roles.
Shrinking the graph by a factor of two horizontally can be achieved by the following changes:
- quadruple a
- double b and d
Similarly, vertically changing b, c and e.
You can shift the graph leftwards by amount M by the following changes:
- change d to d + 2aM
- change e to e + bM
- change f to f + aM^2 + dM
Rotations are more complex again.
Alternatively, we can look at the shape of the conic: circle, ellipse, parabola, hyperbola or just a pair of straight lines. Each of these corresponds to certain relationships between the coefficients.

 

[jimalex]

Haruspex

The math I had (precalc and calc, a long time ago) showed us the general form, but then only really worked with the conics, as you pointed out in the end of your last message. What kind of math book do I need to better understand the general form?

thanks for helping me.

 

[jimalex]

and, yes, ... = 0. you're right, the = 1 was what i was playing with in wolfram alpha when i wrote my message yesterday.

 

[haruspex]

The math I had (precalc and calc, a long time ago) showed us the general form, but then only really worked with the conics, as you pointed out in the end of your last message.

Not sure I understand the question. All quadratics in two variables are conics.
You understand that conics are those curves that can be produced by intersecting a (double-ended) cone with a plane, right? You can see why such intersections can be represented by quadratic equations. Less obvious is why every quadratic equation produces a conic.

 

[jimalex]

yes, sir, i remember that the conics are cases of the general equation. rather than learning the significance of the variables and coefficients in the general form, i was shown how to derive the different conic sections from their definitions and then we studied the transformations of that, for example,

the parabola, being the set of points equidistant from the fucus and the directorix, would be derived by setting the line from the focus to the x,y point equal to the line from the point to the directerix, make some substitutions and simplifications and viola! we had x^2 = 4py. then, some generalizing and rewriting and it's the ax^2 + bx + c = y. the we learned what happens if a is changed, b is changed and so on. just i never learned much about the general form itself. i wondered if understanding all those elements would lead to some kind of unified understanding of all the different sections.

my apologies if i don't make much sense... ignorance is obvious, ain't it

 

[haruspex]

i wondered if understanding all those elements would lead to some kind of unified understanding of all the different sections.

I don't think so. As you'll have seen from my earlier answer, the type of conic results from fairly subtle interplays between the coefficients. I think these statements are true:
- if the xy and either x^2 or y^2 term is missing, you know it's a parabola, but that only spots parabolas oriented to a major axis.
- if the x^2 and y^2 have the same coefficient (including sign), and there's no xy term then it's a circle (and all circles look like that)
- if there's no constant term it passes through the origin
- if x^2 and y^2 are nonzero with the same sign then it's a circle or an ellipse
- if x^2 and y^2 are nonzero with opposite sign then it's a parabola or hyperbola

 

[jimalex]

Thanks for your time.

What I am taking away from this is your lists of observations, the understanding that it is more practical to deal with the general form by boiling it down to a specific section, and that trying to learn to transform the general form itself directly is just too complicated to be an efficient way to work with second degree problems.

thanks again.

 

[micromass]

I don't think so. As you'll have seen from my earlier answer, the type of conic results from fairly subtle interplays between the coefficients. I think these statements are true:
- if the xy and either x^2 or y^2 term is missing, you know it's a parabola, but that only spots parabolas oriented to a major axis.
- if the x^2 and y^2 have the same coefficient (including sign), and there's no xy term then it's a circle (and all circles look like that)
- if there's no constant term it passes through the origin
- if x^2 and y^2 are nonzero with the same sign then it's a circle or an ellipse
- if x^2 and y^2 are nonzero with opposite sign then it's a parabola or hyperbola

A complete classification of quadratic equations is possible from looking at the coefficients only. For example, take the equation

$$ax^2+bxy+cy^2+dx+ey+f=0$$

Let $\delta=b^2-4ac$. The the equation is parabola if $\delta=0$, it is a hyperbola if $\delta>0$ and an ellipse otherwise.

There is also a test to see whether the equation is degenerate (= union of two lines) or not. To see this, define

$$\Delta=det \left(\begin{array}{ccc} a & b/2 & c/2\\ b/2 & c & e/2\\ d/2 & e/2 & f\end{array} \right)$$

If this is zero, then the equation is degenerate. It is nondegenerate otherwise. (note that the 2x2-submatrix in the upper left corner defines $\delta$)
Furthermore, to put an equation in standard form, you simply need to diagonalize the above matrix.

More information is on http://en.wikipedia.org/wiki/Conic_section.

 

[HallsofIvy]

Another way to look at this- the type of conic is determined by the quadratic terms only and we can write those as a matrix multiplication:
$$ax^2+ bx+ c= \begin{bmatrix}x & y \end{bmatrix} \begin{bmatrix}a & b/2 \\ b/2 & c\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}$$
and look for the eigenvalues and eigenvectors. If the eigenvalues are the same, we have a circle, if different but of the same sign, an ellipse, if of different sign a hyperbola, and if one eigenvalue is 0, a parabola.

 

[Vargo]

A geometric interpretation is that by translating your coordinates you can eliminate the linear coefficients d,e. Then by rotating coordinates, you can transform the quadratic part to something that looks like
Au^2+Bv^2+ C = 0.
Here u,v are the rotated coordinates. The values A,B are the eigenvalues Halls is talking about and the rotation to get the coordinates u,v is achieved by changing basis using the associated eigenvectors. The fact that they are orthogonal (and thus produce a rotation of coordinates) follows from the fact that the matrix ( a b // b c) is symmetric.

 

[jimalex]

I appreciate all the response in the last day!

The stuff you all are showing me is not in any book I have, which would be precalc/calculus stuff mostly. Eigenvalues and eigenvectors are from linear algebra, aren't they?

can y'all point me towards a book or two so i can study up on this?

thanks again.

 

[Vargo]

Are you asking for something that explains quadratic forms or are you asking for a linear algebra book recommendation?

I know that Q.forms are discussed in Apostol's Calculus, vol. 2. In linear algebra books, Q. forms are buried quite deep I think (if they are in there at all). In Hoffman and Kunze for example, I think it was the last chapter. Also, most linear algebra books eschew the use of calculus in favor of purely algebraic methods of proof. Personally, I find those algebraic methods harder to understand. I mean, I could read through them and follow the logic, but I couldn't repeat it two days later. On the other hand, more than a decade after reading, I still remember the explanation found in Apostol.

 

[cool_jessica]

Ax2 + Bxy + Cy2 + Dx + Ey + F = 1. The term Diophantine Equation means that the solutions (x, y) should be integer numbers. For example, the equation 4y2 - 20y + 25 = 0 has solutions given by the horizontal line y = 2.5, but since 2.5 is not an integer number, we will say that the equation has no solutions.