Question

Determine horizontal and vertical asymptotes of the graph of this function. Then sketch and explain steps to the graph.

f(x)= (4x)/(x^2 +2)

Answer 1

Let the given function is -

For Vertical Asymptotes -

In order to find vertical asymptotes , we equate the denominator of the above function with zero . So, we have

or ,

,

Which is imaginary , which means it does not exist in real .

Therefore , we will have no vertical asymptotes for the given function f(x).

For Horizontal Asymptotes -

In order to find horizontal asymptotes, we must remember the following rules , which say

• If both polynomials in numerator and denominator are of same degree , find the asymptote by diving the coefficient of highest degree terms .
• If the polynomial in numerator is of lower degree than the polynomial in denominator , then ( y = 0 ), i.e., x - axis is the horizontal asymptotes .
• If the polynomial in numerator is of higher degree than the denominator , then there is no horizontal asymptote of the function .

Here , in our case , clearly , the polynomial in numerator i.e., 4x is of lower degree (of 1 )than the polynomial in denominator , which is of degree 2 ,

Therefore , Horizontal Asymptote of the given function is x - axis , i.e., y = 0 .

Now Tracing the Curve :

Given function is -

symmetry: Since if we replace x by (-x) and y by (-y) in the above equation , the equation remains unchanged , i.e.,

which implies ,

Therefore , the above function is symmetric in the opposite quadrants .

passing through origin : The given curve passes through origin , if it does not contain any constant term .

So , our curve does not contain any constant term , therefore it passes through origin .

asymptotes ; we have already found out the asymptotes of the function , which have , no vertical asymptote and horizontal asymptote as x-axis i.e., y = 0.

table: Now , we will find some points lying the curve in the following table :

x	0	1	-1	2	-2
y	0	4/3	-4/3	4/3	-4/3

Thus , using above information we will graph the function which comes out be , as given below :