4.6 Parametric equations: graphs  (Page 4/4)

Graph on the domain $\left[-\pi ,0\right],$ where $a=5$ and $b=4$ , and include the orientation.

If $a$ is 1 more than $b,$ describe the effect the values of $a$ and $b$ have on the graph of the parametric equations.

Describe the graph if $a=100$ and $b=99.$

What happens if $b$ is 1 more than $a?$ Describe the graph.

If the parametric equations $x(t)=t^2$ and $y\left(t\right)=6-3t$ have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

$x(t)=-t^2$ and $y\left(t\right)$ is linear

$y(t)=t^2$ and $x\left(t\right)$ is linear

$y(t)=-t^2$ and $x\left(t\right)$ is linear

Write the parametric equations of a circle with center $\left(0,0\right),$ radius 5, and a counterclockwise orientation.

Write the parametric equations of an ellipse with center $\left(0,0\right),$ major axis of length 10, minor axis of length 6, and a counterclockwise orientation.

$a=1,b=2$

$a=2,b=1$

$a=3,b=3$

$a=5,b=5$

$a=2,b=5$

$a=5,b=2$

Graph all three sets of parametric equations on the domain $[0,2\pi ].$

Graph all three sets of parametric equations on the domain $\left[0,4\pi \right].$

Graph all three sets of parametric equations on the domain $\left[-4\pi ,6\pi \right].$

The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along?

Explain the effect on the graph of the parametric equation when we switched $\sin t$ and $\cos t$ .

Explain the effect on the graph of the parametric equation when we changed the domain.

An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation $y\left(t\right)=-16t^2+20t$ , while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation $y\left(t\right)=-16t^2+10t+5.$ Write parametric equations for the ball’s position, and then eliminate time to write height as a function of horizontal position.

Find parametric equations that model the problem situation.

Find all possible values of $x$ that represent the situation.

When will the dart hit the ground?

Find the maximum height of the dart.

At what time will the dart reach maximum height?

An epicycloid: $\{\begin{array}{l}x(t)=14\cos t-\cos (14t)\hfill \\ y(t)=14\sin t+\sin (14t)\hfill \end{array}$ on the domain $[0,2\pi ]$ .

A hypocycloid: $\{\begin{array}{l}x(t)=6\sin t+2\sin (6t)\hfill \\ y(t)=6\cos t-2\cos (6t)\hfill \end{array}$ on the domain $[0,2\pi ]$ .

A hypotrochoid: $\{\begin{array}{l}x(t)=2\sin t+5\cos (6t)\hfill \\ y(t)=5\cos t-2\sin (6t)\hfill \end{array}$ on the domain $[0,2\pi ]$ .

A rose: $\{\begin{array}{l}x(t)=5\sin (2t)\sin t\hfill \\ y(t)=5\sin (2t)\cos t\hfill \end{array}$ on the domain $[0,2\pi ]$ .