Determine the vertical asymptotes and holes for the graph of the equation below. Show all your work.
X-1/ x^2+5x-6
To find the vertical asymptotes and holes, we need to examine the equation and determine the values of x that make the denominator equal to zero.
The equation is: (x - 1) / (x^2 + 5x - 6)
To find the vertical asymptotes, we set the denominator equal to zero and solve for x:
x^2 + 5x - 6 = 0
We can factor the quadratic equation to solve for x:
(x + 6)(x - 1) = 0
Setting each factor equal to zero gives us two possible values for x:
x + 6 = 0 => x = -6
x - 1 = 0 => x = 1
Therefore, the vertical asymptotes occur at x = -6 and x = 1.
To determine if there are any holes in the graph, we look for common factors in the numerator and denominator. In this equation, there is one common factor of (x - 1).
By canceling the common factor, we can simplify the equation to:
(x - 1) / (x + 6)
Since cancelling the common factor removes the value x = 1 from the denominator, there is a hole at x = 1.
Therefore, the vertical asymptotes are x = -6 and x = 1, and there is a hole at x = 1.