Question 1

Given the parametric equations $$x = 2 - 5t$$ and $$y = 3 + 3t$$ for $$0 \leq t \leq 1$$, which of the following represents the Cartesian equation after eliminating the parameter $$t$$?

Accepted Answer

$$y = -\frac{3}{5}x + \frac{21}{5}$$

Question 2

A baseball is hit at a height of 4 ft above the ground with an initial velocity of 120 ft/sec at a $$30^\circ$$ angle. The path is modeled by $$x = 120t\cos(30^\circ)$$ and $$y = -16t^2$ + 120t\sin(30^\circ) + 4. $$How high is the ball when it reaches a horizontal distance of 350 ft?

Accepted Answer

Between 23 ft and 28 ft

Question 3

Which of the following best describes a parametric curve?

Accepted Answer

A set of ordered pairs $$(x, y)$$ where $$x = f(t)$$ and $$y = g(t)$$ for $$t in $$some interval $$I$$

Question 4

Given the parametric equations $$x = t^2$ and y = 5 - t$, what is the Cartesian equation after eliminating the parameter?

Accepted Answer

$$y = 5 - \sqrt{x}$$

Question 5

Which of the following is an example of an implicitly defined function?

Accepted Answer

$$2x + 3y = 6$$

Question 6

A line segment has endpoints $$A(2, 3)$$ and $$B(-3, 6). $$Which of the following is a correct parametrization for the segment?

Accepted Answer

$$x = 2 - 5t,\ y = 3 + 3t$$ for $$0 \leq t \leq 1$$

Question 7

If a curve is defined parametrically by $$x = t^2$ - 4$$ and $$y = 2t + 1$$, what is the slope $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

$$\frac{1}{2}$$

Question 8

Which of the following statements about parametric equations is TRUE?

Accepted Answer

They can model real-world motion such as the path of a projectile.

Question 9

Given the parametric equations $$x = 3t + 1$$ and $$y = 2t - 2$$, what is the equation of the line in slope-intercept form?

Accepted Answer

$$y = \frac{2}{3}x - \frac{8}{3}$$

Question 10

A curve is defined by $$x = t^2$ and y$$ = $$t^3. $$What is the average rate of change of $$y$$ with respect to $$x as$$ $$t$$ goes from $$1 to$$ $$2$$?

Accepted Answer

$$\frac{7}{3}$$

Question 11

Which of the following best describes the advantage of writing a function explicitly (e.g., $$y = f(x)) $$rather than implicitly (e.g., $$Ax + By = C$$)?

Accepted Answer

It clearly defines the dependent variable as a function of the independent variable.

Question 12

If $$x = \cos t$$ and $$y = \sin t$$ for $$0 \leq t \leq 2\pi$$, what is the Cartesian equation of the curve?

Accepted Answer

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

Question 13

A projectile is launched from a height of 4 ft with an initial velocity of 120 ft/sec at a $$30^\circ$$ angle. If a fence is 30 ft high and 350 ft away, what happens to the ball?

Accepted Answer

The ball does not clear the fence and will hit the wall.

Question 14

Given the parametric equations $$x = t + 1$$ and $$y = 2t - 3$$, what is the slope of the line represented by these equations?

Accepted Answer

$$2$$

Question 15

Which of the following is NOT a valid reason to use parametric equations?

Accepted Answer

To simplify all algebraic equations.

Parametric and Implicit Functions in Precalculus: Concepts, Applications, and Examples

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