From Math Images
Really nice job with the edits. Your overall presentation is exceptionally clear.
If it's possible to create a graph for the "Finding the Focus and Directrix" section, great, but if it doesn't work, no problem. One thing you could do to simplify this would be to graph a parabola, then add a dot and a dashed line using Microsoft word (Brendan could show you how to do this). Then, instead of actually putting mathematical expressions in the drawing itself, simply refer to the drawings in your text.
It would be really nice to add a section relating the material from the more mathematical section to the idea of a parabola as representing the trajectory of a thrown object, such as an interpretation of a, b, and c (standard form), and h and k (vertex form) for a thrown object (not so hard). This is totally optional, though.
The remaining changes I would say are necessary are all really small:
- At the beginning of the standard form section, maybe replace "The most commonly used..." with "Any vertically oriented parabola can be written using the equation [insert equation]. Also, the graph of any equation in this form will be a parabola." The problem with "most commonly used" is that it doesn't make it clear that a vertical parabola can always be described this way.
- To go from standard to vertex form, it requires that we factor the equation. In some cases, the equation as it is will be factorable. Factoring in general is only very indirectly helpful in writing a quadratic in standard form. For instance x^2 + 7x + 10 = (x+5)(x+2), but it takes a bit of doing to get from there to vertex form, so maybe delete these two sentences or replace them with something vague, like "sometimes there are short cuts, but in general...
- Two small issues with "Graphs of parabolas are often oriented vertically, so that the parabola opens upward or downward, though they can also open sideways. In this case, the equations for the parabola will be written oriented vertically to maintain consistency. To see horizontal parabolas, the x and y can be switched."
- Parabolas can actually also open diagonally. The equations just get much more unpleasant.
- The sentence "In this case..." is a really helpful sentence, the wording is just a bit confusing. By "In this case" do you mean "In the equations below" and when you say "the equations... will be written oriented vertically" I think you mean the equations represented vertically oriented parabolas, while the wording implies that the equations themselves are oriented vertically.
I believe I addressed most of the changes... I will try to make an image for the focus/directrix section, but I am not very confident in my drawing/graphing abilities.
This page is really nice. I agree with Anna that it would help to hide each of the derivation steps.
The one other change that I think is critical (but it's much easier!) is to make all the y's in the more mathematical description lower case.
A couple of other small ideas, with most important changes at the beginning (but none of them are critical):
- At the beginning of your "Standard Form" section, qualify what types of parabolas this equation can apply to. Something like, "Graphs of parabolas are often oriented vertically, so that the parabola opens upward or downward. In this case, the equation of the parabola can be written as..."
- Similarly, in the derivation, explicitly note that this derivation is assuming a horizontal directrix, which is not necessary for parabolas in general.
- An image for the section "Finding the Focus and Directrix" would be extremely helpful, though I can imagine it might be a big pain to find or generate.
- In the Basic Description, eliminate the sentence that reads, "To the left we can see a plane slicing the lower cone so that it goes through the circular base." This sentence isn't really necessary, and eliminating it removes the need to talk about a finite cone two sentences after you specifically make the point that cones are infinite.
I know it'd tedious to do, but I do think it would look better if you hid each of the derivation steps (I think one of Brendan's pages has a good example of this) of the derivation instead of the whole thing.
As someone clicking on it, my thoughts are "oh, let's see a derivation" *click* "GAHHH, that's so much math at once! How can I make it go away again?" Basically, having all of it appear at once is scary, and even my eyes glaze over it a bit. That's the only thing I'd change, though.
I took out the word "general" in the mouse over for "base" to make that sentence more clear.
Can you hide the contents of the algebraic steps, while still keeping the summaries above? Like "now we square both sides" [show equation] ?
"but also in MANY OTHER FIELDS SUCH AS physics and engineering.'
I wouldn't say that the parabola is "used in" those fields, I'd say it arises in them.
I'd force all of your equations to be the same size in your derivation. (you can do this by using \left( and \right) for your parentheses for just one set of parentheses.)
Your mouse over on the word "vertex" should be reworded.
Change "like with standard form" to "as with standard form"
Are you going to finish up that last section here? Or do you think you might change it into an idea for the future?
Also, I bet that there are some really cool optics pictures and applets that illustrate the things you're talking about. You might want to google "parabolic mirror" or "parabolic lens" applets or images to get some more ideas and see new stuff. That would also bring you back to your initial idea that a parabola comes up in other fields.