How do you graph #f(x)=8/(x(x+2))# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Avery Alexander · Accepted Answer

holes: a value that causes both the numerator and denominator to equal zero. there are no holes in this rational function. vertical asymptotes: it's a line #-># set the denominator of the rational function equal to #0#: #x(x+2) = 0#
vertical asymptotes: #x = 0, x = -2# horizontal asymptotes: 
the following are the rules for solving horizontal asymptotes:
let m be the degree of the numerator
let n be the degree of the denominator if m > n, then there is no horizontal asymptote if m = n, then the horizontal asymptote is dividing the coefficients of the numerator and denominator if m < n, then the horizontal asymptote is #y = 0#. As we can see in our rational function, the denominator has a larger degree of #x#. So the horizontal asymptote is #y = 0#. x-ints: x-intercepts are the top of the rational function. Since the numerator just says #8#, that means that there are no x-ints. y-ints: y-intercepts are when you plug in #0# to the function:
#8/(0(0+2))#
#8/0 -> # undefined, so there are no y-ints. Hope this helps!

Jameson Allen · Answer

undefined To graph the function f(x) = 8/(x(x+2)), we can start by identifying the holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

Holes: 
To find the holes, we need to determine the values of x that make the denominator zero. In this case, x(x+2) = 0 when x = 0 or x = -2. Therefore, we have two potential holes at x = 0 and x = -2.

Vertical Asymptotes:
Vertical asymptotes occur when the denominator of a rational function equals zero, but the numerator does not. In this case, the denominator x(x+2) equals zero at x = 0 and x = -2. Therefore, we have two vertical asymptotes at x = 0 and x = -2.

Horizontal Asymptotes:
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 0 and the degree of the denominator is 2. Since the degree of the denominator is greater, the horizontal asymptote is y = 0.

X-intercepts:
To find the x-intercepts, we set the numerator equal to zero and solve for x. In this case, the numerator is 8. Setting it equal to zero, we find that there are no x-intercepts.

Y-intercept:
To find the y-intercept, we substitute x = 0 into the function. Plugging in x = 0, we get f(0) = 8/(0(0+2)) = 8/0 = undefined. Therefore, there is no y-intercept.

To summarize:
- Holes: Potential holes at x = 0 and x = -2.
- Vertical Asymptotes: Vertical asymptotes at x = 0 and x = -2.
- Horizontal Asymptote: Horizontal asymptote at y = 0.
- X-intercepts: No x-intercepts.
- Y-intercept: No y-intercept.