Suggestions for Writing Problem #1

I want to make sure that everyone is clear about what is expected of them with this first writing assignment.

The problem is to describe the graph of the equation |x²-y|=|y²-x|. This description should include supporting statements: that is, it should be a description together with an explanation of how you know that description to be true.

The idea is to practice getting mathematical ideas expressed clearly in writing, using words and mathematical symbols. Here is something to try: imagine giving your description over the phone; every word you say into the phone should be written on the paper. Since we are writing mathematics, some of the words may be expressible using mathematical symbols. For instance, "<" can stand for the words "is less than".

Useful words and phrases include "thus", "so", "as a result of this", "because", "consider", "if", etc.

Take a look at example 4 on page 63 of Stewart to see the sort of writing that I expect. Notice that there are two equivalent equations in a row after "Then we solve this equation for x...": no explanation is needed there because the author assumes that the reader understands the algebraic steps needed to solve the equation. Simple algebraic steps generally need no supporting statements.

Examples 9 and 10 on page 96 of Stewart are also good examples.

General comments and suggestions about writing problems:

• Whenever you are writing mathematics, you want to convey to the reader not just the results of your work, but enough details of the process of your work so that the reader can verify your result. It may help you to write as though you are making a supporting argument for your results: you want the reader to believe your result is correct. If there is a step in your argument that might cause a reader to ask "why is that true?", that's a place you may want to add more explanation.
• How we write mathematics depends a lot on who our intended audience is. For this class, please write in a way that a general student taking calculus would understand. So, you don't have to give an explanation of the quadratic formula, for example.
• Using mathematical notation is okay! In fact, it is encouraged. Please don't write out things like "x squared minus one is equal to y squared minus 1" when you can just write x²-1=y²-1. Use words to convey things that cannot be conveyed with mathematical notation.
  
  Similarly, you don't need to include words of explanation for every little bit of algebra or arithmetic.
• Be sure you understand the distinction between equations and functions. For instance, the expression |x²-1|=|y²-1| is an equation, not a function. It may help to remember that a function is a relationship between two variables (call them y and x) so that there is at most one value of y for each value of x.
  
  An equation is any algebraic expression with an equals sign (=) in it.
  
  We often refer to a generic chunk of algebra as an "expression".
• When working with absolute value, keep in mind that the equations y=|x| and y = ± x are not equivalent. Notice that the first one defines a functional relationship between y and x (i.e., there is one value of y for each x) while the other does not. Think about what the graphs of each equation looks like (hint: they are different).
• Sometimes in mathematics, one can use a right-pointing arrow to stand for the word "implies" or "which implies". This notation is often used incorrectly, so I recommend not using it. Please use phrases like "which implies", or "thus" instead.
• A sentence starts with a capital letter.