Question

How do you graph `f(x)=(x^2-4)/(x^2+4)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

See graph below

Explanation:

First factor the numerator and denominator to see if there are any holes (there aren't).

Then set the numerator equal to zero to find the roots (x intercepts) of the equation

`x^2-4=0`
`(x+2)(x-2) = 0`
`x= +-2`

Then set the denominator equal to zero to find the vertical asymptotes (there aren't any)

To find the y intercept, set all the xs equal to 0
`y = ((0)^2-4)/((0)^2+4)`
`y=-1`

Finally, because the highest exponent in the numerator is equal to the highest exponent in the denominator (2) to find the end behavior (horizontal asymptotes), divide the coefficient of the highest power term in the numerator by the coefficient of the highest power term in the denominator

`1/1 = 1`

Now with the information we can graph
x int = `+-2`
y int = -1
end behavior:
`lim f(x) = 1`
`x->+-oo`

graph{(x^2-4)/(x^2+4) [-8.89, 8.89, -4.444, 4.445]}