list the intercepts & test for symmentry y=4x/(x^2+4)  

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embizze | High School Teacher | (Level 1) Educator Emeritus

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List the intercepts and test for symmetry `y=(4x)/(x^2+4)`

(1) If x=0y=0 so the y-intercept is 0

(2) If y=0 x=0 so the x-intercept is 0

`(4x)/(x^2+4)=0` The denominator is positive, so `4x=0==>x=0`

(3) If we replace x with (-x) we get `y=(-4x)/((-x)^2+4)=(-4x)/(x^2+4)` which is not the same so there is no symmetry about the y-axis.

If we replace y with (-y) we get `-y=(4x)/(x^2+4)==>y=(-4x)/(x^2+4)` Again this is not the same so there is no symmetry about the x-axis.

If we replace x with (-x), y with (-y) we get:

`-y=(4(-x))/((-x)^2+4)==>-y=(-4x)/(x^2+4)` Multiplying both sides by (-1) we get `y=(4x)/(x^2+4)` which is the same. The function is symmetri about the origin.

The graph:

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