Question

test for symmetry with respect to each axis and to the origin. y=x^3+x

Answer 1

first I should see if its symmetric with the origin..i believe you change the x and y values signs.. can you show that first?

Answer 2

There are two ways to do this -- visually (using a grapher) or algebraically. Are you supposed to do one or the other?

Answer 3

I am supposed to show my work, so yeah algebraically

Answer 4

\[(-y) = (-x)^3 + (-x) \\ -y = -x - x\]

Compare that with what you get originally

Answer 5

So far, to check if symmetric on origin, I have y=x^3+x, then I made it -y=(-x)^3 - x

Answer 6

that doesn't match, so it isn't symmetric with the origin?

Answer 7

Ok, that makes sense. Are you allowed to verify your work or use a grapher as you are solving the problem? (It will help you make sure your answers are correct.)

There are two main tests:

1) Symmetric about origin
Everywhere you see an x, replace it with a y and vice versa.

2) Symmetric about the y-axis
Everywhere you see an x , replace it with -x

3) If the function was symmetric about the y axis... it wouldn't be a function (wouldn't be a function of x anyway). (why is this?)

Answer 8

I have to show it on paper. I already checked origin, and found it isn't symmetric about it,right? Now can you help me find if it is symmetric with x and y? :)

Answer 9

So if symmetric with y axis? would I have: y= -x - x after simplification?

Answer 10

@schmidtdancer yup, you showed it's not symmetric about the origin. My first step was actually finding if it was symmetric about the line y = x.

Answer 11

y = x^3 + x

replace x with -x

y = (-x)^3 + (-x)
y = -x -x
y = -2x

Clearly this is not the same.

Answer 12

ok, thats what I had. so not symmetric about origin OR y-axis. Now, lastly, do i check if symmetric about x-axis?

Answer 13

By the way there are LOTS and LOTS of worked examples here: http://tutorial.math.lamar.edu/Classes/Alg/Symmetry.aspx

They might be useful to reference against more complex problems.

Answer 14

ok.

Answer 15

y = (-x)^3 + (-x)
y = (-1)^3(x)^3 - x notice that we are raising -x to the third. This is just negative one to the third times x to the third.

y = -x^3 -x

I think it might be too early for me to be on open study. I feel I'm making too many errors. :(

Answer 16

So we are wrong??

Answer 17

Oh that was for symmetry on y-axis.... But it is still no? right,

Answer 18

Here's how it breaks down.

1) Symmetric about the origin = YES
2) Symmetric about x axis = NO
3) Symmetric about y axis = NO

I'm going to write up the reasons why and double-check them. I apologize for any confusion. :( I'm usually more on the ball but I just woke up.

Answer 19

No prob! Are u sure about those?

Answer 20

Hey sorry I'm back.

Answer 21

Symmetric about origin?
replace x with -x
replace y with -y
If the equation is the same, then YES. Otherwise, NO.

1) Origin
y = x^3 + x
(-y) = (-x)^3 + (-x) replacement of variables
-y = -x^3 - x note that a negative raised to an ODD power is always negative.
y = x^3 +x dividing through by -1 gives us the original equation.
YES

2) x-axis
replace y with -y
If the equation is the same, then YES. Otherwise, NO.

y = x^3 + x
-y = x^3 + x replacement of variable
the equations are NOT the same, you can't get rid of the negative in front of the y.
NO

3) y-axis
replace x with -x

y = x^3 + x
y = (-x)^3 + (-x) replacement of variable
y = -x^3 - x note that a negative raised to an ODD power is always negative.
the equations are NOT the same, you can't get rid of the negative in front of the xs.
NO

Answer 22

I usually don't like to type up full replies like that with such explicit answers, but in this case I felt it was needed because I was being confusing in the beginning. Hope this helps, and please check out http://tutorial.math.lamar.edu/Classes/Alg/Symmetry.aspx for even more examples. :)