Find the intercepts, symmetries, domain, range and (vertical and horizontal) asymptotes. Also Sketch the graph:
y-intecept: This is the point (0,y) on the graph, so:
So the y-intercept is (0,0).
x-intercepts: These are the points (x,0) on the graph, so:
So the x-intercepts are (0,0), (0, -2), and (0, 2).
Parity: What is y(-x)?
Since y(-x) = -y(x) we know that y is an odd function.
Reflecting y over the x-axis takes y -> -y:
Since this is not equal to the original y, there is no reflection symmetry.
Reflecting over the y-axis takes x -> -x
As this is the same as the parity symmetry, we already know that y(-x) is not equal to y(x). So there is no reflection symmetry.
This takes (x,y) -> (-x,-y):
This gives the original function back again, so y(x) is symmetric under inversion.
We look for values of x that makes the function undefined. There are no fractions, square roots, log functions, etc. so the domain is unrestricted. Thus the domain of y is .
We look for all possible values the function can take on its domain. The function is continuous. Note that y -> for x -> and y -> for x -> . So the range of y is .
Horizontal asymptotes: These occur when . In this case so there are no horizontal asymptotes.
Vertical asymptotes: These occur when the denominator of a function becomes 0. There are no denominators in this function so there are no vertical asymptotes.
Slant asymptotes: These occur when . In this case so there are no slant asymptotes.
You should be able to sketch this yourself. However for completeness I have attached one at that bottom of this post.
isnt that (0,0), (-2,0) & (2,0)?So the x-intercepts are (0,0), (0, -2), and (0, 2).
And thank you very much for the help mr top quark!
So you need to find the coordinates of the equation :
since its symmetrical w/ respect to the origin ryt?