Graph y=6/(x^2)

Problem

y=6/(x2)

Solution

1. Identify the domain and check for vertical asymptotes. Since the denominator is x2 the function is undefined at x=0 There is a vertical asymptote at x=0

2. Determine the horizontal asymptote by examining the limit as x approaches infinity. As x→∞ or x→−∞ the value of y approaches 0 There is a horizontal asymptote at y=0

3. Check for symmetry by substituting −x for x Since ƒ*(−x)=6/((−x)2)=6/(x2)=ƒ(x) the function is even and symmetric about the y-axis.

4. Analyze the sign of the function. Because x2 is always positive for all x≠0 the value of y is always positive. The graph remains entirely above the x-axis.

5. Plot key points to determine the shape. For x=1 y=6 For x=2 y=1.5 For x=3 y=2/3 Due to symmetry, the points (−1,6) (−2,1.5) and (−3,2/3) are also on the graph.

6. Sketch the curve starting near the vertical asymptote x=0 where y goes to positive infinity, passing through the plotted points, and approaching the horizontal asymptote y=0 as |x| increases.

Final Answer

y=6/(x2)

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