Q: Which of the following is a correct parameterization for the line passing through $$P_0$$ = ($$x_0$$, $$y_0$$)$$ and P_1$ = (x_1$, y_1$)$$, with $$t = 0 at$$ $$P_0$$ and $$t = 1 at$$ $$P_1$$?

$$x = x_0$ + (x_1$ - x_0$)t, \\ y = y_0$ + (y_1$ - y_0$)t$$

Q: Given the parametric equations $$x = a \\cos t$$, $$y = b \\sin t$$, $$0 \\leq t \\leq 2\\pi$$, $$a \u003e b \u003e 0$$, what is the Cartesian equation of the curve?

$$\\frac{x^2$}{a^2$} + \\frac{y^2$}{b^2$} = 1$$

Q: Consider the parametric equations $$x = \\frac{1}{1 + t^2$}$$, $$y = \\frac{t}{1 + t^2$}$$, $$t \\in (-\\infty, \\infty). $$What is the center and radius of the curve described?

Center at $$\\left(\\frac{1}{2}, 0\\right)$$, radius $$\\frac{1}{2}$$

Q: A circle of radius 1 centered at the origin can be parameterized in multiple ways. Which of the following parameterizations traces the circle TWICE as $$t$$ goes from $$0 to$$ $$4\\pi$$?

$$x = \\cos t, \\ y = \\sin t, \\ t \\in [0, 4\\pi]$$

Question 1

Which of the following statements best describes a parametric curve in the plane?

Accepted Answer

A set of points $$(x, y)$$ such that $$x = f(t)$$ and $$y = g(t)$$ for $$t in $$some interval $$I.$$

Question 2

Given the parametric equations $$x = \cos t$$, $$y = \sin t$$, $$t \in [0, \pi]$$, what geometric object is described by these equations?

Accepted Answer

The upper half of the unit circle centered at the origin.

Question 3

Eliminate the parameter $$t$$ from the equations $$x = t^2$ - 1$$, $$y = t + 1. $$What is the resulting Cartesian equation?

Accepted Answer

$$x = y^2$ - 2y$$

Question 4

Which of the following is a correct parameterization for the line passing through $$P_0$$ = ($$x_0$$, $$y_0$$)$$ and P_1$ = (x_1$, y_1$)$$, with $$t = 0 at$$ $$P_0$$ and $$t = 1 at$$ $$P_1$$?

Accepted Answer

$$x = x_0$ + (x_1$ - x_0$)t, \ y = y_0$ + (y_1$ - y_0$)t$$

Question 5

Given the parametric equations $$x = a \cos t$$, $$y = b \sin t$$, $$0 \leq t \leq 2\pi$$, $$a \u003e b \u003e 0$$, what is the Cartesian equation of the curve?

Accepted Answer

$$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$

Question 6

Which of the following statements about parameterizations is TRUE?

Accepted Answer

A curve can have many different parameterizations.

Question 7

Given $$x = \cos \sqrt{t}$$, $$y = \sin \sqrt{t}$$, $$t \in [0, 4\pi^2$]$$, what curve is traced out in the $$xy-$$plane?

Accepted Answer

A circle of radius 1 traced twice.

Question 8

Consider the parametric equations $$x = \frac{1}{1 + t^2$}$$, $$y = \frac{t}{1 + t^2$}$$, $$t \in (-\infty, \infty). $$What is the center and radius of the curve described?

Accepted Answer

Center at $$\left(\frac{1}{2}, 0\right)$$, radius $$\frac{1}{2}$$

Question 9

Suppose a particle moves along the curve $$x = t^2$ - 1$$, $$y = t + 1$$ for $$t \in (-\infty, \infty). $$Which of the following best describes the direction of motion as $$t$$ increases?

Accepted Answer

The particle traces a parabola opening to the right, moving upwards as $$t$$ increases.

Question 10

Which of the following is NOT a valid way to eliminate the parameter $$t$$ from $$x = a \cos t$$, $$y = b \sin t$$?

Accepted Answer

Set $$x = y$$ and solve for $$t.$$

Question 11

A circle of radius 1 centered at the origin can be parameterized in multiple ways. Which of the following parameterizations traces the circle TWICE as $$t$$ goes from $$0 to$$ $$4\pi$$?

Accepted Answer

$$x = \cos t, \ y = \sin t, \ t \in [0, 4\pi]$$

Question 12

Which of the following real-world scenarios can be best modeled using parametric equations?

Accepted Answer

The position of a car moving along a winding road as a function of time.

Question 13

Given the parametric equations $$x = (x_1$ - x_0$)t + x_0$, y$$ = ($$y_1$$ - $$y_0)t$$ + $$y_0$$, for $$t \in [0, 1]$$, what is the geometric meaning of $$t = 0.5$$?

Accepted Answer

The point halfway between $$P_0$$ and $$P_1.$$

Question 14

If a curve is parameterized by $$x = f(t)$$, $$y = g(t)$$, and $$t$$ increases, what does the direction of increasing $$t$$ represent?

Accepted Answer

The orientation or direction along the curve.

Question 15

Suppose $$x = \cos t$$, $$y = \sin t$$, $$t \in [0, 2\pi]. At $$which value of $$t$$ does the point $$(0, 1)$$ occur?

Accepted Answer

$$t = \frac{\pi}{2}$$

Question 16

Which of the following is a valid parameterization for the ellipse centered at the origin with semi-axes $$a$$ and $$b$$?

Accepted Answer

$$x = a \cos t, \ y = b \sin t, \ t \in [0, 2\pi]$$

Parametric Curves: Concepts, Examples, and Applications

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