Describe the vertical asymptote(s) and hole(s) for the graph of y = .(x+2)(x+4)/(x+4)(x+1)
To determine the vertical asymptotes and holes of the given rational function, we need to examine the factors in the numerator and denominator.
The numerator is (x + 2)(x + 4), and the denominator is (x + 4)(x + 1).
To find vertical asymptotes, we need to look for values that would make the denominator equal to zero, since division by zero is undefined. In this case, there are two potential vertical asymptotes: when x = -4 and x = -1.
Now let's check if there are any holes in the graph. A hole occurs when a factor in the numerator and denominator cancel out, resulting in a removable discontinuity.
In this case, we see that there is a common factor of (x + 4) in both the numerator and denominator. By canceling out this factor, we are left with y = (x + 2)/(x + 1). This means that there is a hole at x = -4.
To summarize:
- There is a vertical asymptote at x = -4.
- There is another vertical asymptote at x = -1.
- There is a hole at x = -4.