Calculus/Calculus

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Expert: - 8/15/2010


Question
Hello!  I am working on my AP Calculus summer assignment that is due Tuesday and am having trouble on one of the questions.  

Check for symmetry with respect to both axes and to the origin.
((x^2)y)-(x^2)+4y = 0

I hope you can help me!
Thanks so much!!

Answer

Plot
Hello Hannah,

Regarding the symmetry of graphs...
1. If replacing "x" with "-x" does not change the equation,
.  then the graph is also unchanged and it is symmetric about
.  the y-axis.  Hence, an even function.
2. If replacing "y" with "-y" does not change the equation,
.  then the graph is also unchanged and it is symmetric about
.  the x-axis.
3. If replacing "x" with "-x" AND "y" with "-y" does not change
.  the equation, then the graph is also unchanged and it is
.  symmetric about the origin.  Hence, an odd function.

So...for ((x^2)y)-(x^2)+4y = 0
1. You can see replacing "x" with "-x" produces no change,
.  because of the even powers of "x"...so it is symmetric about
.  the y-axis.
2. You can check that replacing "y" with "-y" yields a different
.  equation, so no symmetry about the x-axis.
3. Similarly, for this case as well...no symmetry about (0,0).

I have attached an image of the graph!

I hope this helps...and good luck with your AP exam!  :-)
(I used to be a grader for the AP Calculus exams!)

TTYL, Abe

    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    CommentThanks so much! Your answer really helped!


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Abe Mantell

Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook

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