매개변수 방정식과 곡선

[T] Sketch the curve given by parametric equations $\begin{array}{c}x=\cosh (t) \\ y=\sinh (t),\end{array}$ where $-2\le t\le 2.$

T/F: When sketching the graph of parametric equations, the $x$ and $y$ values are found separately, then plotted together.

The direction in which a graph is “moving” is called the _____ of the graph.

An equation written as $y=f(x)$ is written in _____ form.

Create parametric equations $x=f(t)$, $y=g(t)$ and sketch their graph. Explain any interesting features of your graph based on the functions $f$ and $g$.

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^2+t$, $y=1-t^2$, $-3\leq t\leq 3$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=1$, $y=5\sin t$, $-\pi/2\leq t\leq \pi/2$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^2$, $y=2$, $-2\leq t\leq 2$

In Exercises 5–8, sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph.

$x=t^3-t+3$, $y=t^2+1$, $-2\leq t\leq 2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=t^3-2t^2$, $y=t^2$, $-2\leq t \leq 3$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=1/t$, $y=\sin t$, $0< t \leq 10$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=3\cos t$, $y=5\sin t$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=3\cos t+2$, $y=5\sin t+3$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t$, $y=\cos(2t)$, $0\leq t \leq \pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t$, $y=\sin(2t)$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=2\sec t$, $y=3\tan t$, $-\pi/2< t < \pi/2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cosh t$, $y=\sinh t$, $-2\leq t \leq 2$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t+\frac14\cos(8t)$, $y=\sin t+\frac14\sin(8t)$, $0\leq t \leq 2\pi$

In Exercises 9–18, sketch the graph of the given parametric equations; using a graphing utility is advisable. Be sure to indicate the orientation of the graph.

$x=\cos t+\frac14\sin(8t)$, $y=\sin t+\frac14\cos(8t)$, $0\leq t \leq 2\pi$

In Exercises 19–20, four sets of parametric equations are given. Describe how their graphs are similar and different. Be sure to discuss orientation and ranges.

(a) $x=t$ $y=t^2$, $-\infty< t< \infty$ (b) $x=\sin t$ $y=\sin^2t$, $-\infty< t< \infty$ (c) $x=e^t$ $y=e^{2t}$, $-\infty< t< \infty$ (d) $x=-t$ $y=t^2$, $-\infty< t< \infty$