Question 1

Given the parametric equations $$x(t) = 2t - 1$$ and $$y(t) = t^2$ - 2$$, which of the following is the correct rectangular equation after eliminating the parameter?

Accepted Answer

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

Question 2

If $$x(t) = t - 1$$ and $$y(t) = t^2$ - 2$$, what is the rectangular equation relating $$x$$ and $$y$$?

Accepted Answer

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

Question 3

Given $$x(t) = t - 3$$ and $$y(t) = t^2$, which of the following is the correct rectangular equation?

Accepted Answer

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

Question 4

Given $$x = 2 + \cos t$$ and $$y = 1 - \sin t$$, eliminate the parameter to find the rectangular equation.

Accepted Answer

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

Question 5

For the parametric equations $$x = 3\cos t$$, $$y = 2\sin t$$, $$0 \leq t \leq 2\pi$$, what is the rectangular equation?

Accepted Answer

$$\frac{x^2$}{9} + \frac{y^2$}{4} = 1$$

Question 6

Which of the following best describes the process of eliminating the parameter from parametric equations?

Accepted Answer

Solving one equation for $$t$$ and substituting into the other equation

Question 7

Given $$x = 2\cos t$$ and $$y = 3\sin t$$, which identity is used to eliminate the parameter?

Accepted Answer

$$\cos^2$ t + \sin^2$ t = 1$$

Question 8

Suppose a particle moves along a path described by $$x = t^2$, y = t + 1$. What is the rectangular equation for the path?

Accepted Answer

$$y = \sqrt{x} + 1$$

Question 9

Given $$x = 1 + 2\cos t$$ and $$y = 2\sin t$$, what is the rectangular equation after eliminating the parameter?

Accepted Answer

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

Question 10

A Ferris wheel is modeled by $$x = 5\cos t$$, $$y = 5\sin t. $$What is the rectangular equation for the path of a seat?

Accepted Answer

$$x^2$$ + $$y^2 = 25$

Question 11

Given $$x = t^2$ + 1$$ and $$y = t - 2$$, what is the rectangular equation relating $$x$$ and $$y$$?

Accepted Answer

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

Question 12

If $$x = 2\cos t$$ and $$y = 2\sin t$$, what geometric shape does the rectangular equation represent?

Accepted Answer

A circle centered at the origin with radius 2

Question 13

Given $$x = t$$ and $$y = t^2$ - 4t + 3$$, what is the rectangular equation?

Accepted Answer

$$y = x^2$ - 4x + 3$$

Question 14

A satellite moves in a path described by $$x = 4\cos t$$, $$y = 3\sin t. $$What is the rectangular equation for its orbit?

Accepted Answer

$$\frac{x^2$}{16} + \frac{y^2$}{9} = 1$$

Question 15

Given $$x = t + 2$$ and $$y = t^2$, which of the following is the correct rectangular equation?

Accepted Answer

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

Question 16

If $$x = 2\cos t$$ and $$y = 3\sin t$$, which of the following is the correct rectangular equation?

Accepted Answer

$$\frac{x^2$}{4} + \frac{y^2$}{9} = 1$$

Question 17

Given $$x = t^2$ and y$$ = $$t^3$$, what is the rectangular equation relating $$x$$ and $$y$$?

Accepted Answer

Eliminating the Parameter in Parametric and Trigonometric Equations

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