Trace the curve.... y^2(x^2-9)=x^4.... by stating all the properties used in tracing.....
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The curve does not exist for -3<x<3 because there can be no solutions for y when x^2<9.

Because of the presence of x^2 and y^2 the curve is essentially reflected in all quadrants. If we examine QI only we have the same patterns reflected in the other quadrants. x=-3 and x=3 are asymptotes.

As x gets larger, the constant 9 becomes more insignificant and y^2 becomes x^2, in other words, the curve in QI has an asymptote y=x, which is reflected in all the other quadrants. So we know the curve is trapped between the vertical asymptote and the line y=x.

(2ydy/dx)(x^2-9)+2y^2x=4x^3 is the derivative. When dy/dx=0, there is a turning point. y^2=2x^2. That is, x^4/(x^2-9)=2x^2. x^4=2x^4-18x^2; x^2=2x^2-18; x^2=18, x=3sqrt(2) and y^2=36, so y=6. In QI then, there is a turning point at (3sqrt(2),6). This is reflected in each quadrant. This is a minimum (in QI) because the curve is trapped between the above asymptotes. There is enough information now to trace the curve(s). The asymptotes resemble a cross intersected by vertical lines. The asymptote in QI and QIV has been drawn in to illustrate how close the curve comes to it.

 

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What are the properties of tracing?? (just for my knowledge)

The properties are the properties of the function given:

  • Where is the function defined (domain and range)? These give a clue to the shape.
  • Are there any extrema? Maximum and minimum points are useful in tracing.
  • Are there any asymptotes? These help to define the constraints for the curve. They can also be drawn on to the graph to show where the curve approaches them.
  • Is the curve constinuou? Are there any holes or discontinuities?
  • What is the behaviour for large values of the variables? (This can help to define asymptotes.)
  • Is the curve reflected (usually even exponents give rise to the same quantities for positive and negative values)?

All these properties define the behaviour of a curve and help to trace (sketch) it.

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