10.7 Parametric Equations: Graphs

What are two methods used to graph parametric equations?

Why are some graphs drawn with arrows?

Why are parametric graphs important in understanding projectile motion?

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

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

Describe the graph of the set of parametric equations.

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

Describe the graph of the set of parametric equations.

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

Use a graphing utility to graph the given parametric equations.

1. $\begin{cases} x \left(t\right) = \cos t - 1 \\ y \left(t\right) = \sin t + t \end{cases}$
2. $\begin{cases} x \left(t\right) = \cos t + t \\ y \left(t\right) = \sin t - 1 \end{cases}$
3. $\begin{cases} x \left(t\right) = t - \sin t \\ y \left(t\right) = \cos t - 1 \end{cases}$

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

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 $y \left(t\right) = - 16 t^{2} + 20 t$, 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.

Use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time $t .$ Neglect air resistance.

Find parametric equations that model the problem situation.

Use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time $t .$ Neglect air resistance.

When will the dart hit the ground?

Use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time $t .$ Neglect air resistance.

At what time will the dart reach maximum height?