Graph the function `f(x)=(2x^2)/(x^2+3)` :
A rational function will have a vertical asymptote if the function has been written with no common factors in the numerator and denominator and the denominator has value 0 for some value(s) of x.
Here the denominator `x^2+3` is positive for all x, so there are no vertical asymptotes.
Since the degree of the numerator and denominator is the same, the function will have a horizontal asymptote. The asymptote is `y=a_n/b_n` where `a_n,b_n` are the leading coefficients of the numerator and denominator respectively, when written in standard form.
Here the horizontal asymptote is y=2.
The graph will have an x-intercept if the numerator is zero and the denominator is nonzero at some value of x.
Here the numerator is zero at x=0, and the denominator is nonzero, so there is an x-intercept at x=0. The y-intercept is also at x=0, so the y-intercept is 0.
The numerator is nonnegative for all values of x, and the denominator is positive for all values of x, so the graph is nonnegative for all values of x.
Trying some test values we find (0,0),(1,.5),(3,1.5) are on the graph. The graph is symmetric about the y-axis so (-1,.5) and (-3,1.5) are also on the graph.
Note the horizontal asymptote at y=2.