Q: A projectile is launched so that its position at time $$t is $$given by $$x = v_0$ t \cos \theta$$, $$y = v_0$ t \sin \theta - \frac{1}{2}gt^2$. What is the Cartesian equation for the projectile's path?

$$y = x \tan \theta - \frac{g}{2v_0^2$ \cos^2$ \theta} x^2$

Question 1

Which of the following best describes a parametric equation?

Accepted Answer

An equation where $$x$$ and $$y$$ are both given as functions of a third variable, called a parameter.

Question 2

Given the parametric equations $$x = 	an t$$, $$y = 	an^2$ t$$, for $$0 \u003c t \u003c \frac{\pi}{2}$$, which of the following is the Cartesian equation after eliminating the parameter?

Accepted Answer

$$y = x^2$

Question 3

A particle moves along a curve so that its position at time $$t is $$given by $$x = 2t$$, $$y = t^2$. Which of the following best describes the path traced by the particle?

Accepted Answer

A parabola opening upwards

Question 4

Which of the following is a correct set of parametric equations for a circle of radius $$r$$ centered at $$(h, k)$$?

Accepted Answer

$$x = h + r \cos t$$, $$y = k + r \sin t$$

Question 5

Suppose a curve is given by $$x = \sin t$$, $$y = \cos t$$ for $$0 \leq t \leq \pi. $$What is the geometric shape of the curve and which portion is traced?

Accepted Answer

Upper half of the unit circle, traced from $$(0,1) to$$ $$(0,-1)$$

Question 6

Which of the following statements about parametric equations is TRUE?

Accepted Answer

Different parametric equations can represent the same curve.

Question 7

Given the parametric equations $$x = 3t + 2$$, $$y = 2t - 1$$, what is the Cartesian equation relating $$x$$ and $$y$$?

Accepted Answer

$$y = \frac{2}{3}x - \frac{7}{3}$$

Question 8

A bicycle wheel of radius $$a$$ rolls along a straight line without slipping. The path traced by a point on the rim is given by $$x(t) = a t - a \sin t$$, $$y(t) = a - a \cos t. $$What is the name of this curve?

Accepted Answer

Cycloid

Question 9

Which of the following is a correct set of parametric equations for the line passing through points $$(5,0)$$ and $$(0,-1)$$?

Accepted Answer

$$x = 5 - 5t$$, $$y = -t$$, \quad 0 \leq t \leq 1$

Question 10

If a curve is given by $$x = f(t)$$, $$y = g(t)$$, which of the following is NOT a valid reason for using parametric equations?

Accepted Answer

To make all curves linear

Question 11

Given $$x = \sin t$$, $$y = \sin^2$ t$$, eliminate the parameter to find the Cartesian equation.

Accepted Answer

$$y = x^2$

Question 12

Which of the following best describes the process of 'eliminating the parameter'?

Accepted Answer

Converting a pair of parametric equations to a single equation relating $$x$$ and $$y$$

Question 13

A projectile is launched so that its position at time $$t is $$given by $$x = v_0$ t \cos 	heta$$, $$y = v_0$ t \sin 	heta - \frac{1}{2}gt^2$. What is the Cartesian equation for the projectile's path?

Accepted Answer

$$y = x 	an 	heta - \frac{g}{2v_0^2$ \cos^2$ 	heta} x^2$

Question 14

Which of the following parametric equations describes a circle of radius $$2$$ centered at $$(0,0)$$ traced counterclockwise?

Accepted Answer

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

Question 15

Which of the following is a possible parametric representation for the parabola $$y = x^2$?

Accepted Answer

$$x = t$$, $$y = t^2$

Question 16

A curve is given by $$x = f(t)$$, $$y = g(t)$$, $$a \leq t \leq b. $$What does the interval $$a \leq t \leq b$$ represent?

Accepted Answer

The range of the parameter $$t$$ over which the curve is traced

Question 17

Which of the following is a true statement about the direction in which a parametric curve is traced?

Accepted Answer

The direction is determined by increasing values of the parameter.

Question 18

Given $$x = 2t - 1$$, $$y = 3t + 4$$, for $$0 \leq t \leq 2$$, what are the coordinates of the endpoints of the curve?

Accepted Answer

$$(x_1$, y_1$) = (-1, 4)$$ and $$(x_2$, y_2$) = (3, 10)$$

Question 19

Which of the following is a real-world application of parametric equations?

Accepted Answer

Describing the path of a planet orbiting the sun

Section 1.1 - Parametric Equations

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