Question 1

Which of the following correctly describes a parametric equation?

Accepted Answer

An equation where $$x$$ and $$y$$ are both defined as functions of a third variable, usually $$t$$, such that $$x = f(t)$$ and $$y = g(t).$$

Question 2

Given the parametric equations $$x = 2t - 4$$, $$y = 3 + t^2$, eliminate the parameter t$ to write y$ as a function of x$.

Accepted Answer

$$y = 3 + \left( \dfrac{x + 4}{2} \$$right)^2$$

Question 3

Which of the following is a correct parametrization for the line $$y = \dfrac{2}{3}x - 7$$?

Accepted Answer

$$x = t, \quad y = \dfrac{2}{3}t - 7$$

Question 4

Find the parametric equations for the line passing through the points $$(2, -3)$$ and $$(-5, 2).$$

Accepted Answer

$$x = 2 + (-7)t, \quad y = -3 + 5t$$

Question 5

A circle is centered at $$(-3, 2)$$ with radius $$5. $$Which of the following gives its parametric equations?

Accepted Answer

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

Question 6

Given the ellipse $$\dfrac{(x-2)^2$}{9} + \dfrac{(y+1)^2$}{16} = 1$$, which of the following is a correct parametrization?

Accepted Answer

$$x = 2 + 3\cos t, \quad y = -1 + 4\sin t$$

Question 7

A projectile is launched such that its position at time $$t is $$given by $$x = v_0$ t \cos 	heta$$, $$y = v_0$ t \sin 	heta - \dfrac{1}{2}gt^2$. What type of curve does its path represent?

Accepted Answer

A parabola

Question 8

Which of the following is NOT a valid reason to use parametric equations in real-world modeling?

Accepted Answer

To avoid using variables in equations

Question 9

Suppose a car moves along a path described by $$x = 4t$$, $$y = 2t + 1. At$$ $$t = 3$$, what are the coordinates of the car?

Accepted Answer

$$(12, 7)$$

Question 10

Given $$x = 2t - 4$$, $$y = 3 + t^2$, what is the direction of motion as t$ increases?

Accepted Answer

Right and upward

Question 11

Which of the following parametric equations describes a circle of radius $$r$$ centered at $$(h, k)$$?

Accepted Answer

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

Question 12

A parametric curve is given by $$x = t^2$, y$$ = $$t^3. $$Which of the following is the Cartesian equation relating $$y$$ and $$x$$?

Accepted Answer

$$y = x$$^{3/2}$$

Question 13

Which of the following parametric equations describes an ellipse centered at $$(h, k)$$ with semi-axes $$a$$ and $$b$$?

Accepted Answer

$$x = h + a\cos t, \quad y = k + b\sin t$$

Question 14

A cyclist rides along a circular path described by $$x = 10\cos t$$, $$y = 10\sin t. $$What is the radius of the path?

Accepted Answer

$$10$$

Question 15

Given the parametric equations $$x = 3t + 2$$, $$y = -2t + 5$$, what is the slope of the line represented?

Accepted Answer

$$-\dfrac{2}{3}$$

Question 16

Which of the following is the correct way to eliminate the parameter $$t$$ from $$x = 2t - 4$$, $$y = 3 + t^2$?

Accepted Answer

Solve $$x = 2t - 4$$ for $$t$$, then substitute into $$y = 3 + t^2$

Question 17

A parametric curve is given by $$x = t$$, $$y = t^2$. Which of the following best describes the graph?

Accepted Answer

A parabola opening upward

Question 18

Which of the following parametric equations would describe a line segment between $$(1, 2)$$ and $$(4, 8) as$$ $$t$$ goes from $$0 to$$ $$1$$?

Accepted Answer

$$x = 1 + 3t, \quad y = 2 + 6t$$

Question 19

A medical imaging device traces a path described by $$x = 5\cos t$$, $$y = 2\sin t. $$What shape does the device trace?

Accepted Answer

An ellipse

Study Notes: Parametric Equations (Section 10.1)

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