Question 1

Given the parametric equations $$x = t^2$ - 4$$, $$y = \frac{1}{2} t$$, for $$t \in [-2, 3]$$, which of the following is the correct Cartesian equation relating $$x$$ and $$y$$?

Accepted Answer

$$x = 4y^2$ - 4$$

Question 2

For the parametric equations $$x = e$$^{t}$$ + 1$$, $$y = e$$^{2t}$$, which of the following is the correct Cartesian equation?

Accepted Answer

$$y = (x - 1)^2$

Question 3

Given $$x = \sin(t)$$ and $$y = \csc(t)$$ for $$t \in (0, \pi)$$, which of the following is the correct Cartesian equation?

Accepted Answer

$$x y = 1$$

Question 4

Which of the following is the correct parametric equation for a circle with center at $$(-3, 2)$$ and radius $$4$$?

Accepted Answer

$$x = -3 + 4 \cos t, \quad y = 2 + 4 \sin t$$

Question 5

Suppose a particle moves along the path $$x = t^2$ - 4$$, $$y = \frac{1}{2} t$$ for $$t \in [-2, 3]. At $$what value of $$t$$ does the particle cross the $$y-$$axis?

Accepted Answer

$$t = 2$$

Question 6

Given the parametric equations $$x = \sin t$$, $$y = \csc t$$, for $$t \in (0, \pi)$$, which of the following points is on the curve?

Accepted Answer

$$(\frac{\sqrt{2}}{2}, \sqrt{2})$$

Question 7

Which of the following parametric equations describes an ellipse centered at $$(-3, 2)$$ with vertices at $$(-3, 7)$$ and $$(-3, -3)$$, and passing through $$(1, 2)$$ and $$(7, 2)$$?

Accepted Answer

$$x = -3 + 4 \cos t, \quad y = 2 + 5 \sin t$$

Question 8

A circle is described by the parametric equations $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t. $$What is the equation of the circle in Cartesian form?

Accepted Answer

$$(x + 3)^2$ + (y - 2)^2$ = 16$$

Question 9

Given the parametric equations $$x = e$$^{t}$$ + 1$$, $$y = e$$^{2t}$$, what is the value of y$ when x = 2$?

Accepted Answer

$$y = 1$$

Question 10

Suppose a particle moves along the ellipse $$x = -3 + 4 \cos t$$, $$y = 2 + 5 \sin t. At $$what value(s) of $$t$$ does the particle reach the point $$(1, 2)$$?

Accepted Answer

$$t = 0$$

Question 11

Which of the following best describes the geometric shape traced by the parametric equations $$x = e$$^{t}$$ + 1$$, $$y = e$$^{2t}$$ as t$ varies over all real numbers?

Accepted Answer

A rightward-opening parabola

Question 12

Given $$x = t^2$ - 4$$, $$y = \frac{1}{2} t$$, for $$t \in [-2, 3]$$, what is the range of $$y$$?

Accepted Answer

$$[-1, 1.5]$$

Question 13

If $$x = \sin t$$ and $$y = \csc t$$, for $$t \in (0, \pi)$$, which of the following statements is true?

Accepted Answer

As $$x$$ approaches $$0$$, $$y$$ approaches infinity.

Question 14

Which of the following is the correct Cartesian equation for the ellipse with center $$(-3, 2)$$, vertices at $$(-3, 7)$$ and $$(-3, -3)$$, and passing through $$(1, 2)$$ and $$(7, 2)$$?

Accepted Answer

$$\frac{(x + 3)^2$}{16} + \frac{(y - 2)^2$}{25} = 1$$

Question 15

A circle with center $$(-3, 2)$$ and radius $$4 is $$described by $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t. $$What is the length of the path traced by one revolution?

Accepted Answer

$$8\pi$$

Parametric Equations and Cartesian Conversion: Study Notes for Calculus II

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