Graphing Techniques - Finding Possible y-values of a Curve
Graphing Techniques - Finding Possible y-values of a Curve
TGM Original Questions
The curve \(C\) has the equation \(y=\frac{5}{x-1}-\frac{1}{x+1}\).
- Find the coordinates of any points of intersection with the axes.
- State the equations of the asymptotes.
- Without using a calculator, find the set of values that \(y\) can take.
- Sketch \(C\), making clear the relevant features of the curve.
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Answer:(i) \(\left(-\dfrac32,0\right)\) and \((0,-6)\) (ii) \(x=-1\), \(x=1\), and \(y=0\) (iii) \(y\leq-3-\sqrt5\) or \(y\geq-3+\sqrt5\) (iv) local minimum \(\left(\dfrac{-3-\sqrt5}{2},-3+\sqrt5\right)\) and local maximum \(\left(\dfrac{-3+\sqrt5}{2},-3-\sqrt5\right)\), with sketch as shown
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