Question 1

Which of the following best describes a parametric curve?

Accepted Answer

A graph of ordered pairs $$(x, y)$$ where $$x$$ and $$y$$ are both functions of a third variable $$t$$ called the parameter.

Question 2

Given the parametric equations $$x = 5 - t^2$ and y = 2t$, eliminate the parameter t$ to find the Cartesian equation.

Accepted Answer

$$x = 5 - \left(\frac{y}{2}\$$right)^2$$

Question 3

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

Accepted Answer

Between 23.04 ft and 27.76 ft

Question 4

Which of the following is a vector-valued function for the parametric equations $$x = t^2$, y = 2t + 1$?

Accepted Answer

$$\vec{r}(t) = \langle t^2$, 2t + 1 \rangle$$

Question 5

Given the parametric equations $$x = 2t + 1$$, $$y = 4t - 3$$, what is the slope of the line represented by these equations?

Accepted Answer

2

Question 6

Julia walks along a horizontal line with her position given by $$x = 0.1t^3$ - 2t^2$ + 9$$, $$y = 4. At $$what time does she first change direction?

Accepted Answer

At approximately $$t = 4.4$$ seconds

Question 7

Which of the following is an explicit definition of a linear function?

Accepted Answer

$$y = mx + b$$

Question 8

Given the parametric equations $$x = t^2$, y$$ = $$t^3$$, what is the average rate of change of $$y$$ with respect to $$x on $$the interval $$t \in [0, 2]$$?

Accepted Answer

$$2$$

Question 9

Which of the following best describes an implicitly defined function?

Accepted Answer

A function defined by an equation involving both $$x$$ and $$y$$, such as $$Ax + By = C$$

Question 10

Find a parametrization for the line segment with endpoints $$A(1, 2)$$ and $$B(5, 8)$$ for $$0 \leq t \leq 1.$$

Accepted Answer

$$x = 1 + 4t,\ y = 2 + 6t$$

Question 11

If a particle moves along the curve $$x = \cos t$$, $$y = \sin t$$ for $$0 \leq t \leq 2\pi$$, what is the geometric path traced by the particle?

Accepted Answer

A circle of radius 1 centered at the origin

Question 12

Given the parametric equations $$x = t^2$ - 4$$, $$y = 2t$$, what is the vertex of the parabola after eliminating the parameter?

Accepted Answer

$$( -4, 0 )$$

Question 13

Which of the following real-world scenarios can be modeled using parametric equations?

Accepted Answer

The flight path of a golf ball

Question 14

Given $$x = 3t + 2$$, $$y = -t + 5$$, eliminate the parameter and write the equation in terms of $$x$$ and $$y.$$

Accepted Answer

$$y = -\frac{1}{3}x + \frac{11}{3}$$

Question 15

If $$x = t^2$ and y$$ = $$t^3$$, what is the relationship between $$x$$ and $$y$$ after eliminating the parameter?

Accepted Answer

$$y^2$$ = $$x^3$$

Question 16

Which of the following statements about the average rate of change for parametric curves is correct?

Accepted Answer

It is calculated as $$\frac{y_2$ - y_1$}{x_2$ - x_1$}$$ using the corresponding $$t$$ values.

Question 17

A line is defined implicitly by $$2x - 3y = 6. $$What is the explicit form of this line?

Accepted Answer

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

Question 18

Which of the following is NOT a benefit of using parametric equations to model motion?

Accepted Answer

They always produce linear graphs.

Parametric and Vector-Valued Functions, Modeling Motion, and Implicit Functions in Precalculus

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