Question 1

Which of the following best describes a parametric equation for a plane curve?

Accepted Answer

$$x = f(t),\ y = g(t)$$, where $$t is a $$parameter and $$f$$ and $$g$$ are continuous on an interval $$I$$

Question 2

Given the parametric equations $$x = t^2$, y = 2t + 3$ for t$ in $$[-3, 3]$$, what is the equivalent rectangular equation?

Accepted Answer

$$x = \frac{1}{4}(y - 3)^2$

Question 3

For the parametric equations $$x = 2t + 5$$, $$y = \sqrt{4 - t^2$}$$, for $$t in$$ $$[-2, 2]$$, which of the following is the correct rectangular equation?

Accepted Answer

$$\frac{y^2$}{4} + \frac{(x - 5)^2$}{16} = 1$$

Question 4

A projectile is launched at a $$45^\circ$$ angle with initial speed $$v_0. $$Its position is given by $$x = \frac{v_0$ \sqrt{2}}{2} t$$, $$y = \frac{v_0$ \sqrt{2}}{2} t - 16t^2$. What is the rectangular equation for its path?

Accepted Answer

$$y = x - \frac{32}{v_0$^2} x^2$

Question 5

Which of the following statements about parametric equations is TRUE?

Accepted Answer

Parametric representations of a curve are not always unique.

Question 6

Given $$x = t^2$, y$$ = $$t^2$$ for $$t in$$ $$(-\infty, \infty)$$, what is the equivalent rectangular equation and its domain?

Accepted Answer

$$y = x$$, $$x \geq 0$$

Question 7

Which of the following is a valid parametric representation for the parabola $$y = (x - 2)^2$ + 1$$?

Accepted Answer

$$x = t,\ y = (t - 2)^2$ + 1$$

Question 8

A parametric curve is defined by $$x = t^2$, y = 2t + 3$ for t$ in $$[-3, 3]$$. What is the range of x$ and y$?

Accepted Answer

$$x \in [0, 9],\ y \in [-3, 9]$$

Question 9

If a projectile is launched with $$v_0 = 128$ ft/s at a 45$$^\circ$$ angle, what is the coefficient of x^2$ in the rectangular equation for its path?

Accepted Answer

$$-\frac{32}{(128)^2$}$$

Question 10

Which real-world scenario can be modeled using parametric equations?

Accepted Answer

The path of a projectile launched at an angle

Question 11

Given $$x = 2t + 5$$, $$y = \sqrt{4 - t^2$}$$ for $$t in$$ $$[-2, 2]$$, what portion of the ellipse does the graph represent?

Accepted Answer

The upper half of the ellipse

Question 12

Which of the following is NOT a correct way to eliminate the parameter $$t$$ from $$x = t^2$, y = 2t + 3$?

Accepted Answer

Set $$t = x$$ and substitute into $$y = 2t + 3$$

Question 13

If $$x = t$$, $$y = f(t)$$ for $$t in$$ $$X$$, what is the domain of the rectangular equation $$y = f(x)$$?

Accepted Answer

The same as the domain $$X of $$the parameter $$t$$

Question 14

A parametric curve is given by $$x = t^2$, y$$ = $$t^2$$ for $$t \geq 0. $$What is the geometric representation of this curve?

Accepted Answer

A ray along $$y = x$$ for $$x \geq 0$$

Question 15

Which of the following is a correct alternative parametric representation for $$y = (x - 2)^2$ + 1$$?

Accepted Answer

$$x = t + 2,\ y = t^2$ + 1$$

Question 16

In the context of projectile motion, what does the parameter $$t$$ represent in the parametric equations for position?

Accepted Answer

Time elapsed since launch

Parametric Equations: Concepts, Graphs, and Applications

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