Chapter 1

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=t^{2}+2 t, \quad y=t+1 $$

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=\cos (t), y=\sin (t),(0,2 \pi] $$

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=2 t+4, y=t-1 $$

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=3-t, y=2 t-3,1.5 \leq t \leq 3 $$

Eliminate the parameter and sketch the graphs. $$ x=2 t^{2}, \quad y=t^{4}+1 $$

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=t^{2}+t, \quad y=t^{2}-1 $$

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=e^{-t}, \quad y=e^{2 t}-1 $$

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=3 \cos t, \quad y=4 \sin t $$

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=\sec t, \quad y=\cos t $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=e^{t}, \quad y=e^{2 t}+1 $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=6 \sin (2 \theta), y=4 \cos (2 \theta) $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=\cos \theta, \quad y=2 \sin (2 \theta) $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=3-2 \cos \theta, \quad y=-5+3 \sin \theta $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=4+2 \cos \theta, \quad y=-1+\sin \theta $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=\sec t, \quad y=\tan t $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=\ln (2 t), \quad y=t^{2} $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=e^{t}, \quad y=e^{2 t} $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=e^{-2 t}, \quad y=e^{3 t} $$

Sketch the parametric equations by eliminating the parameter. Indicate any asymptotes of the graph. $$ x=4 \sec \theta, \quad y=3 \tan \theta $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{2}-1, \quad y=\frac{t}{2} $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\frac{1}{\sqrt{t+1}}, \quad y=\frac{t}{1+t}, t>-1 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=4 \cos \theta, y=3 \sin \theta, t \in(0,2 \pi] $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\cosh t, \quad y=\sinh t $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=2 t-3, \quad y=6 t-7 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{2}, \quad y=t^{3} $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=1+\cos t, \quad y=3-\sin t $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\sqrt{t}, \quad y=2 t+4 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\sec t, \quad y=\tan t, \pi \leq t<\frac{3 \pi}{2} $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=2 \cosh t, \quad y=4 \sinh t $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=\cos (2 t), \quad y=\sin t $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=4 t+3, y=16 t^{2}-9 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{2}, \quad y=2 \ln t, t \geq 1 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{3}, \quad y=3 \ln t, t \geq 1 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. \(x=t^{n}, \quad y=n \ln t, t \geq 1, \quad\) where \(n\) is a natural number

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ \begin{array}{l} x=\ln (5 t) \\ y=\ln \left(t^{2}\right) \text { where } 1 \leq t \leq e \end{array} $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ \begin{array}{l} x=2 \sin (8 t) \\ y=2 \cos (8 t) \end{array} $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ \begin{array}{l} x=\tan t \\ y=\sec ^{2} t-1 \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=3 t+4 \\ y=5 t-2 \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x-4=5 t \\ y+2=t \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=2 t+1 \\ y=t^{2}-3 \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=3 \cos t \\ y=3 \sin t \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=2 \cos (3 t) \\ y=2 \sin (3 t) \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=\cosh t \\ y=\sinh t \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=3 \cos t \\ y=4 \sin t \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=2 \cos (3 t) \\ y=5 \sin (3 t) \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=3 \cosh (4 t) \\ y=4 \sinh (4 t) \end{array} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{array}{l} x=2 \cosh t \\ y=2 \sinh t \end{array} $$

Use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. $$ \begin{array}{l} x=\theta+\sin \theta \\ y=1-\cos \theta \end{array} $$

Use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. $$ \begin{array}{l} x=2 t-2 \sin t \\ y=2-2 \cos t \end{array} $$

Use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. $$ \begin{array}{l} x=t-0.5 \sin t \\ y=1-1.5 \cos t \end{array} $$