How do you graph and find the discontinuities of #R(x) = (x^2 + x - 12)/(x^2 - 4)#?

Question

Julia Adams · Accepted Answer

The graph would appear to be as in the given figure.

The graph has two vertical asymptotes at x^2-4=0, that is at x=2 and x= -2

R(x) can also be written as R(x)= 1 +#(x-8)/(x^2-4)#. Thus it has horizontal asymptote at y=1.
At x=0, y=3

The graph would cross x axis at R(x)=0, that is when #x^2+x-12=0# Or, (x+4)(x-3)=0, at x=-4, 3

The graph would appear like this

Savannah Anderson · Answer

undefined To graph and find the discontinuities of the function R(x) = (x^2 + x - 12)/(x^2 - 4), follow these steps:

1. Factor the numerator and denominator:
   Numerator: x^2 + x - 12 = (x + 4)(x - 3)
   Denominator: x^2 - 4 = (x + 2)(x - 2)

2. Determine the vertical asymptotes by setting the denominator equal to zero and solving for x:
   x + 2 = 0  -->  x = -2
   x - 2 = 0  -->  x = 2

3. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator:
   Since the degree of the numerator (2) is equal to the degree of the denominator (2), there is a horizontal asymptote at y = 1.

4. Find the x-intercepts by setting the numerator equal to zero and solving for x:
   x + 4 = 0  -->  x = -4
   x - 3 = 0  -->  x = 3

5. Determine the y-intercept by plugging in x = 0 into the function:
   R(0) = (0^2 + 0 - 12)/(0^2 - 4) = -12/-4 = 3

6. Plot the vertical asymptotes, horizontal asymptote, x-intercepts, and y-intercept on a graph.

7. Determine the behavior of the function near the vertical asymptotes:
   As x approaches -2 or 2, the function approaches positive or negative infinity, depending on the sign of the numerator.

8. Determine the behavior of the function as x approaches positive or negative infinity:
   As x approaches positive or negative infinity, the function approaches the horizontal asymptote at y = 1.

9. Connect the points on the graph smoothly, considering the behavior near the asymptotes.

This completes the process of graphing and finding the discontinuities of the function R(x) = (x^2 + x - 12)/(x^2 - 4).