91Ó°ÊÓ

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=3 t-1, \quad y=2 t+1 $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=1+\frac{1}{t}, \quad y=t-1 $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sec \theta, \quad y=\cos \theta $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{3}{2+6 \sin \theta}\)

Write the equation for the ellipse rotated \(\pi / 4\) radian clockwise from the ellipse \(r=\frac{5}{5+3 \cos \theta}\).

Conjecture (a) Use a graphing utility to graph the curves represented by the two sets of parametric equations. \(x=4 \cos t \quad x=4 \cos (-t)\) \(y=3 \sin t \quad y=3 \sin (-t)\) (b) Describe the change in the graph when the sign of the parameter is changed. (c) Make a conjecture about the change in the graph of parametric equations when the sign of the parameter is changed. (d) Test your conjecture with another set of parametric equations.

Sketch the strophoid \(r=\sec \theta-2 \cos \theta\) \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Find two different sets of parametric equations for the rectangular equation. $$ y=x^{3} $$

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$

$$ \text { State the definition of a smooth curve } $$