Graph r(x)=6/((x-2)^2)

Problem

r(x)=6/((x−2)2)

Solution

1. Identify the vertical asymptote by finding the value of x that makes the denominator zero.

x−2=0

x=2

2. Determine the horizontal asymptote by observing the degrees of the numerator and denominator. Since the degree of the numerator (0) is less than the degree of the denominator (2), the horizontal asymptote is the x-axis.

y=0

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

r(0)=6/((0−2)2)

r(0)=6/4=1.5

4. Analyze the behavior near the asymptote by checking the sign of the function. Since the denominator (x−2)2 is always positive for x≠2 and the numerator is positive, r(x)>0 for all x in the domain. As x→2 r(x)→∞ from both sides.

5. Plot key points to determine the shape of the curves on either side of the vertical asymptote.

r(1)=6/((1−2)2)=6

r(3)=6/((3−2)2)=6

r(4)=6/((4−2)2)=1.5

6. Sketch the graph using the asymptotes x=2 and y=0 the y-intercept (0,1.5) and the points (1,6) (3,6) and (4,1.5) The graph consists of two branches in the upper half-plane that approach the vertical line x=2

Final Answer

r(x)=6/((x−2)2)

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