91Ó°ÊÓ

Eccentricity Consider an ellipse with eccentricity \(e\) . (a) What are the possible values of \(e ?\) (b) What happens to the graph of the ellipse as \(e\) increases?

Think About It How can two sets of parametric equations represent the same graph but different curves?

Graphing a Conic In Exercises 3 and \(4,\) use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. $$r=\frac{2 e}{1+e \cos \theta}$$

Tangent Lines Consider a curve represented by the parametric equations \(x=f(t)\) and \(y=g(t) .\) When does the graph have horizontal tangent lines? Vertical tangent lines?

Parametric Form of a Polar Equation Explain how to write a polar equation in parametric form.

Arc Length Why does the arc length formula require that the curve not intersect itself on an interval, except possibly at the endpoints?

Adjusting a Domain Consider the parametric equations \(x=\sqrt{t-2}\) and \(y=\frac{1}{2} t+1, \quad t \geq 2.\) What is implied about the domain of the resulting rectangular equation?

Hyperbola Explain how to sketch a hyperbola with a vertical transverse axis.

Graphing a Conic In Exercises 3 and \(4,\) use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. $$r=\frac{2 e}{1-e \sin \theta}$$

Writing In Exercises 5 and \(6,\) consider the polar equation \(r=\frac{4}{1+e \sin \theta}\) Use a graphing utility to graph the equation for \(e=0.1\) \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) .