Graph (x-4)/(x^2-x-6)

Problem

y=(x−4)/(x2−x−6)

Solution

1. Factor the denominator to identify the domain and vertical asymptotes.

y=(x−4)/((x−3)*(x+2))

2. Identify the vertical asymptotes by finding where the denominator equals zero.

x−3=0⇒x=3

x+2=0⇒x=−2

3. Identify the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (2), the asymptote is the x-axis.

y=0

4. Find the x-intercept by setting the numerator equal to zero.

x−4=0⇒x=4

5. Find the y-intercept by evaluating the function at x=0

y=(0−4)/(0−0−6)=(−4)/(−6)=2/3

6. Analyze the behavior of the graph in the intervals created by the asymptotes: (−∞,−2) (−2,3) and (3,∞)

• For x<−2 the function is negative.

• For −2<x<3 the function is positive.

• For 3<x<4 the function is negative.

• For x>4 the function is positive.

Final Answer

y=(x−4)/((x−3)*(x+2))

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