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

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

Question 5

Suppose you are given the parametric equations $$x = t^2$ - 4$$, $$y = \frac{1}{2} t. $$What is the range of $$x as$$ $$t$$ varies from $$-2 to$$ $$3$$?

Accepted Answer

$$x \in [0, 5]$$

Question 6

Given the ellipse centered at $$(-3, 2)$$ with vertices at $$(-3, 7)$$ and $$(-3, -3)$$, which of the following is the correct parametric equation?

Accepted Answer

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

Question 7

A particle moves along the curve $$x = e$$^{t}$$ + 1$$, $$y = e$$^{2t}$$. At t = 0$, what is the slope of the tangent to the curve?

Accepted Answer

$$2$$

Question 8

Which of the following best describes the graph of $$x = \sin t$$, $$y = \csc t$$ for $$t \in (0, \pi)$$?

Accepted Answer

A hyperbola in the $$xy-$$plane

Question 9

If a circle is parameterized as $$x = h + r \cos t$$, $$y = k + r \sin t$$, what is the corresponding Cartesian equation?

Accepted Answer

$$(x - h)^2$ + (y - k)^2$ = r^2$

Question 10

Given the parametric equations $$x = t^2$ - 4$$, $$y = \frac{1}{2} t$$, what is the value of $$y$$ when $$x = 0$$?

Accepted Answer

$$y = 1$$

Question 11

Suppose you want to graph the ellipse with center $$(-3, 2)$$, vertices at $$(-3, 7)$$ and $$(-3, -3)$$, and two other points at $$(1, 2)$$ and $$(-7, 2). $$What are the lengths of the semi-major and semi-minor axes?

Accepted Answer

Semi-major axis: $$5$$, semi-minor axis: $$4$$

Question 12

For the parametric equations $$x = -3 + 2 \cos t$$, $$y = 2 + 5 \sin t$$, what is the range of $$y$$?

Accepted Answer

$$[-3, 7]$$

Question 13

Which of the following points lies on the circle parameterized by $$x = -3 + 4 \cos t$$, $$y = 2 + 4 \sin t$$ for some $$t$$?

Accepted Answer

$$(1, 2)$$

Question 14

If $$x = e$$^{t}$$ + 1$$ and $$y = e$$^{2t}$$, what is $$\frac{dy}{dx}$$ in terms of t$?

Accepted Answer

$$\frac{2e$$^{2t}$$}{e$$^{t}$$}$$

Question 15

Given the parametric equations $$x = t^2$ - 4$$, $$y = \frac{1}{2} t$$, what is the value of $$t$$ when $$y = -1$$?

Accepted Answer

$$t = -2$$

Study Notes: Parametric Equations and Cartesian Conversion

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