Graph (4x^2)/(x^2-2x-3)

Problem

ƒ(x)=(4*x^2)/(x^2−2*x−3)

Solution

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

x^2−2*x−3=(x−3)*(x+1)

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

x−3=0⇒x=3

x+1=0⇒x=−1

3. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both are degree 2, divide the leading coefficients.

y=4/1=4

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

4*x^2=0⇒x=0

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

ƒ(0)=(4*(0)^2)/(0^2−2*(0)−3)=0

6. Analyze the behavior of the graph near the asymptotes and intercepts to sketch the curve. The function is negative between x=−1 and x=3 (except at x=0 and positive elsewhere.

Final Answer

ƒ(x)=(4*x^2)/((x−3)*(x+1))

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