Explain in detailed steps how to sketch y= 1/x^2-4
Determine the values of x for which y is undefined, those would be when the denominator is 0.
Factor the expression to find the roots.
The asymptotes are x=-2 and x=2, thus the domain of the function is x<-2, -2<x<2 and x>2
Determine the range: y`!=` 0, but y approaches 0 when x`->oo` or
x`->-oo` , observe that the graph is symmetric with respect to the x=0 axis because the asymptotes are symmetric.
Determine the value of y when x=0.
Determine the value of y when x is close to 2, x=1.5 and x=-1.5
Thus y=-0.57 when x=1.5 and x=-1.5. Since the value of y at x=0 is higher (-0.25>-0-0.57), the value of y will become more negative as it approaches the asymptotes.
Determine the value of y when x is close to 2 on the other side of the asymptote. Compute y for x=2.999
The value of y is positive and cannot be 0, then the curve has asymptotes x=2 and y=0, same for x=-2 and y=0 on the other side.
Verify using graphing: