To graph a rational function function, we need to determine its asypmtotes.
Its vertical asymptote is the value of x that would give us a zero denominator. To solve, set the denominator equal to zero.
x = 0
Hence, the vertical asymptote is x = 0.
To determine its horizontal asymptote refer to the degree of the numerator and the denominator. The degree of the numerator is zero and the denominator is 2.
Since the degree of the numerator is less than that of the denominator, the horizontal asymptote is y=0.
Next, assign values of x to the left of the vertical asymptote to determine the direction of the graph.
If `x = -1 ` `x = -8`
`y = 4/(-1)^2 = 4` `y=4/(-8)^2=4/64=1/16`
Both y's are postive, this means that at interval `(-oo, 0)` , the graph is located above the horizontal asymptote. As indicated by the values of y, as x approaches the vertical asymptote, the value of y increases. But as x decreases the value of y approcahes the horizontal asymptote.
Then, assign values of x to the right of the vertical asymptote.
If `x = 1 ` `x = 8`
`y=4/x^2=1` ` y = 4/8^2=4/64=1/16`
The values of y are positive too. This indicates that at interval `(0,+oo)` , the graph is located at the top of HA. Also, as the value of x approaches the VA, y increases. And, as the value of x increases, the y approaches the HA.
Hence the graph of `f(x) = 4/x^2` is:
(Note: Yellow - VA , Green - HA, and Blue - graph of `f(x)=4/x^2` )