list the intercept and test for symmentry y= 3x / {x^2 +16}

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embizze's profile pic

Posted on

Given `y=(3x)/(x^2+16):`

(1) The y-intercept is found when x=0 ==> the y-intercept is at 0.

(2) The x-intercept(s) are found when you set y=0 and solve:

`(3x)/(x^2+16)=0` The numerator is always positive, so for the expression on the left to be 0 the numerator must be 0. 3x=0==>x=0. So the x-intercept is at 0.

(3) (a) Test for symmetry about the y-axis -- substitute (-x) for x:

`(3(-x))/((-x)^2+16)=(-3x)/(x^2+16)!=(3x)/(x^2+16)` so the function is not symmetric about the y-axis.

(b) To test for symmetry about the y-axis substitute (-y) for y:

`-y=(3x)/(x^2+16)==> y=(-3x)/(x^2+16)!= (3x)/(x^2+16)` so the function is not symmetric about the x-axis.

(c) To test for symmetry about the origin substitute (-x) for x and (-y) for y:

`-y=(3(-x))/((-x)^2+16)`

`-y=(-3x)/(x^2+16)`

`y=(3x)/(x^2+16)` which is the same as the original, so the function is symmetric about the origin. (It is an odd function)

The graph:

lfryerda's profile pic

Posted on

The x-intercept is found when y=0 and the y-intercept is found when x=0.  For this function, both the x-intercept and the y-intercept are therefore found at the origin (0,0).

To test for symmetry, make the substitution `f(-x)` .  The function is odd if `f(-x)=-f(x)` and even if `f(-x)=f(x)` .

`f(-x)={-x}/{(-x)^2+16}`

`=-x/{x^2+16}`

`=-f(x)`

Therefore the function is odd and the intercept is at (0,0).

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