Graph f(x)=(x^2-4x)/(x^2-9)

Problem

ƒ(x)=(x2−4*x)/(x2−9)

Solution

1. Factor the numerator and denominator to identify zeros and discontinuities.

ƒ(x)=(x*(x−4))/((x−3)*(x+3))

2. Find the x-intercepts by setting the numerator equal to zero.

x(x−4)=0⇒x=0,x=4

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

ƒ(0)=(0−4*(0))/(0−9)=0

4. Identify vertical asymptotes by finding where the denominator is zero (and the numerator is non-zero).

(x−3)*(x+3)=0⇒x=3,x=−3

5. Determine the horizontal asymptote by comparing the degrees of the numerator and denominator.

(lim_x→∞)((x2−4*x)/(x2−9))=1⇒y=1

6. Analyze behavior near the vertical asymptotes and intercepts to sketch the curve.

As *x→−3,ƒ(x)→∞

As *x→−3,ƒ(x)→−∞

As *x→3,ƒ(x)→∞

As *x→3,ƒ(x)→−∞

Final Answer

To graph ƒ(x)=(x2−4*x)/(x2−9) plot the intercepts at (0,0) and (4,0) draw vertical asymptotes at x=−3 and x=3 and a horizontal asymptote at y=1

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