# 8.7 Parametric equations: graphs  (Page 4/4)

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Graph on the domain $\text{\hspace{0.17em}}\left[-\pi ,0\right],\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b=4$ , and include the orientation.

If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is 1 more than $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ describe the effect the values of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ have on the graph of the parametric equations.

Describe the graph if $\text{\hspace{0.17em}}a=100\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b=99.$

There will be 100 back-and-forth motions.

What happens if $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is 1 more than $\text{\hspace{0.17em}}a?\text{\hspace{0.17em}}$ Describe the graph.

If the parametric equations $\text{\hspace{0.17em}}x\left(t\right)={t}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\left(t\right)=6-3t\text{\hspace{0.17em}}$ have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

Take the opposite of the $\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}$ equation.

For the following exercises, describe the graph of the set of parametric equations.

$x\left(t\right)=-{t}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\left(t\right)\text{\hspace{0.17em}}$ is linear

$y\left(t\right)={t}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}$ is linear

The parabola opens up.

$y\left(t\right)=-{t}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\left(t\right)\text{\hspace{0.17em}}$ is linear

Write the parametric equations of a circle with center $\text{\hspace{0.17em}}\left(0,0\right),$ radius 5, and a counterclockwise orientation.

$\left\{\begin{array}{l}x\left(t\right)=5\mathrm{cos}t\\ y\left(t\right)=5\mathrm{sin}t\end{array}$

Write the parametric equations of an ellipse with center $\text{\hspace{0.17em}}\left(0,0\right),$ major axis of length 10, minor axis of length 6, and a counterclockwise orientation.

For the following exercises, use a graphing utility to graph on the window $\text{\hspace{0.17em}}\left[-3,3\right]\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}\left[-3,3\right]\text{\hspace{0.17em}}$ on the domain $\text{\hspace{0.17em}}\left[0,2\pi \right)\text{\hspace{0.17em}}$ for the following values of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b$ , and include the orientation.

$\left\{\begin{array}{l}x\left(t\right)=\mathrm{sin}\left(at\right)\\ y\left(t\right)=\mathrm{sin}\left(bt\right)\end{array}$

$a=1,b=2$

$a=2,b=1$

$a=3,b=3$

$a=5,b=5$

$a=2,b=5$

$a=5,b=2$

## Technology

For the following exercises, look at the graphs that were created by parametric equations of the form $\text{\hspace{0.17em}}\left\{\begin{array}{l}x\left(t\right)=a\text{cos}\left(bt\right)\hfill \\ y\left(t\right)=c\text{sin}\left(dt\right)\hfill \end{array}.\text{\hspace{0.17em}}$ Use the parametric mode on the graphing calculator to find the values of $a,b,c,$ and $d$ to achieve each graph.

$a=4,\text{\hspace{0.17em}}b=3,\text{\hspace{0.17em}}c=6,\text{\hspace{0.17em}}d=1$

$a=4,\text{\hspace{0.17em}}b=2,\text{\hspace{0.17em}}c=3,\text{\hspace{0.17em}}d=3$

For the following exercises, use a graphing utility to graph the given parametric equations.

1. $\left\{\begin{array}{l}x\left(t\right)=\mathrm{cos}t-1\\ y\left(t\right)=\mathrm{sin}t+t\end{array}$
2. $\left\{\begin{array}{l}x\left(t\right)=\mathrm{cos}t+t\\ y\left(t\right)=\mathrm{sin}t-1\end{array}$
3. $\left\{\begin{array}{l}x\left(t\right)=t-\mathrm{sin}t\\ y\left(t\right)=\mathrm{cos}t-1\end{array}$

Graph all three sets of parametric equations on the domain $\text{\hspace{0.17em}}\left[0,\text{\hspace{0.17em}}2\pi \right].$

Graph all three sets of parametric equations on the domain $\text{\hspace{0.17em}}\left[0,4\pi \right].$

Graph all three sets of parametric equations on the domain $\text{\hspace{0.17em}}\left[-4\pi ,6\pi \right].$

The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along?

Explain the effect on the graph of the parametric equation when we switched $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t$ .

The $\text{\hspace{0.17em}}y$ -intercept changes.

Explain the effect on the graph of the parametric equation when we changed the domain.

## Extensions

An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation $\text{\hspace{0.17em}}y\left(t\right)=-16{t}^{2}+20t$ , while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

$y\left(x\right)=-16{\left(\frac{x}{15}\right)}^{2}+20\left(\frac{x}{15}\right)$

A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation $\text{\hspace{0.17em}}y\left(t\right)=-16{t}^{2}+10t+5.\text{}$ Write parametric equations for the ball’s position, and then eliminate time to write height as a function of horizontal position.

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time $\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ Neglect air resistance.

Find parametric equations that model the problem situation.

$\left\{\begin{array}{l}x\left(t\right)=64t\mathrm{cos}\left(52°\right)\\ y\left(t\right)=-16{t}^{2}+64t\mathrm{sin}\left(52°\right)\end{array}$

Find all possible values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that represent the situation.

When will the dart hit the ground?

approximately 3.2 seconds

Find the maximum height of the dart.

At what time will the dart reach maximum height?

1.6 seconds

For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.

An epicycloid: $\text{\hspace{0.17em}}\left\{\begin{array}{l}x\left(t\right)=14\mathrm{cos}\text{\hspace{0.17em}}t-\mathrm{cos}\left(14t\right)\hfill \\ y\left(t\right)=14\mathrm{sin}\text{\hspace{0.17em}}t+\mathrm{sin}\left(14t\right)\hfill \end{array}\text{\hspace{0.17em}}$ on the domain $\text{\hspace{0.17em}}\left[0,2\pi \right]$ .

A hypocycloid: $\left\{\begin{array}{l}x\left(t\right)=6\mathrm{sin}\text{\hspace{0.17em}}t+2\mathrm{sin}\left(6t\right)\hfill \\ y\left(t\right)=6\mathrm{cos}\text{\hspace{0.17em}}t-2\mathrm{cos}\left(6t\right)\hfill \end{array}\text{\hspace{0.17em}}$ on the domain $\text{\hspace{0.17em}}\left[0,2\pi \right]$ .

A hypotrochoid: $\left\{\begin{array}{l}x\left(t\right)=2\mathrm{sin}\text{\hspace{0.17em}}t+5\mathrm{cos}\left(6t\right)\hfill \\ y\left(t\right)=5\mathrm{cos}\text{\hspace{0.17em}}t-2\mathrm{sin}\left(6t\right)\hfill \end{array}\text{\hspace{0.17em}}$ on the domain $\text{\hspace{0.17em}}\left[0,2\pi \right]$ .

A rose: $\text{\hspace{0.17em}}\left\{\begin{array}{l}x\left(t\right)=5\mathrm{sin}\left(2t\right)\mathrm{sin}t\hfill \\ y\left(t\right)=5\mathrm{sin}\left(2t\right)\mathrm{cos}t\hfill \end{array}\text{\hspace{0.17em}}$ on the domain $\text{\hspace{0.17em}}\left[0,2\pi \right]$ .

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations