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## Table of Contents

Quick InstructionsAdd Parametrics content to this page. See specific details on the home page.

## Parametrics (Unit 10) Learning Targets

## Parametric Equations

x- and y- coordinatesof a point on a curve are given asseparate equations in terms of an independent variable(called a parameter).Graphs of parametric equations are often used to

model location and motion over time:x-y graphtells you therelationship between x and ybut does not describe the object's pathparametert isto tell you the effect of an independent variableon x and y## Procedure for graphing parametric equations

When given a t-x equation and a t-y equation1) Sketch the graphs of t-x and t-y equations

2) Eliminate the parameter t and write the equation in rectangular (x-y) form

3) Note any restrictions on t and find the range of x and y

4) Graph the x-y equation

5) Describe the direction of the graph using the t-x and t-y graphs

## Example 1

Click to Show/Hide1) t-x graph:

t-y graph:

2) Eliminate the parameter t and write the rectangular (x-y) form

3) We know from our rational functions unit that since x = 4/(t-2) t cannot equal 2. The t-x graph shows that t cannot equal 2 and x cannot equal 0. The t-y graph shows that since t cannot equal 2, there is a PORD (hole) at (2,3).

4) rectangular (x-y) graph:

5) The t-y graph shows that the y values are always increasing as t increases, so the y values on the x-y graph will also always increase.

As t increases, the x values decrease from ∞ to 0 then from 0 to -∞.

As t increases, the y values increase from -∞ to 3 then from 3 to ∞.

## Example 2

x = 4cos(t) , y = 4sin(t)1) t-x graph:

t-y graph:2) Remember from our Trig Identities Unit that cos^2(t) + sin^2(t) = 1So x^2 = (4^2)cos^2(t)y^2 = (4^2)sin^2(t)(x^2)/16 + (y^2)/16 = 1x^2 + y^2 = 4^23) t = all real #sx: [ -4, 4]y: [ -4, 4]4) x-y graph:5) When t=0, the graph starts at (4, 0) and then goes to (0, 4). Thus, the graph travels counterclockwise.

REMEMBER...**

Different parametrics can produce the same x-y graph / motion!!!The rectangular equation and graph only tell you the total displacement and overall path but do not give the directional changesThe parametric equations and graphs only tell you the effects of an independent variable on the x-positions and y-positions separately but do not show the actual path that is taken by the object.## Extra Resources

An Overview of the Parametrics UnitParametric Equation Examples## Parametric vs Function Mode on a Calculator

functionmode, ay-equationisdefined in terms of xparametricmode, for each parametric equation, there isa x-equation and y-equation. Both aredefined in terms of the parameter t## Sequential vs. Simultaneous Method of Graphing on a Calculator

Sequential- Graphsparametrics one at a timeSimultaneous- Graphsparametrics at the same timeTI-83/84 instruction:press [Mode] and choose the mode you want (SEQUENTIAL or SIMUL)TI-89 instruction:In the Y= editor, press [F1] and under "Graph Order" choose (SEQ or SIMUL)## Visualizing Parametrics on a Calculator - Flaming Beach Ball

current location of the object is surrounded by a circle (beach ball)and leaves behind a trail of the previous locationssimultaneousandnon-simultaneous solutionsof a set of parametric equationsTI-83/84 instruction: press [Y =], to the left of the x-t equation there is a diagonal line, move the cursor to the left twice to hover it, press [ENTER] twice to find the flaming beach ball option (should look like " -o ")TI-89 instruction: In the Y= editor, press [2ND] and then [F1] (to open up menu F6), press [6] to choose "Path"## Graphing Parametrics and Functions on the TI-83/84

http://www.prenhall.com/esm/app/graphing/ti83/Home_Screen/Menu_Keys/mode/mode.html## Extra Resources

Graphical 'Mode' Significancehttp://www.algebralab.org/lessons/lesson.aspx?file=calculator_calculatormodes.xml

## Graphing Parametrics and Functions on the TI-89

https://www.math.lsu.edu/~neal/TI_89/graphing/other_modes/parametric/parametric.html## Simultaneous Solutions and Non-Simultaneous Solutions

occur at theSimultaneous solutionssame placeat thesame timeoccur at theNon-simultaneous Solutionssame placebut atdifferent times## Procedure for finding simultaneous solutions

## Procedure for non-simultaneous solutions

## Example 3

Determine all simultaneous and non-simultaneous solutions for the set of parametric equations:For simultaneous:Steps 1) and 2), set y and x equations equal to each other and solve for t

3) We see that from setting the equations equal to each other, parameters both equal -2. Therefore, there is a simultaneous solution when t= -2.

4) To find the full result/solution, we substitute -2 back into one pair of x y equations (to find where the collision occurs at t= -2):

For non-simultaneous:1) Eliminate parameters to have two y-x equations

2) Solve by combination or elimination (where Y1 = Y2 and X1 = X2)

3) Sub back into equations to solve for corresponding y values

(-4, 0) and ( -3, 0) are all the solutions of the two x-y equations. But (-4,0) as shown above is a simultaneous solution. So only (-3, 0) is a non-simultaneous solution :)

## Example 4

Izzy is riding a ferris wheel and his path and direction are described by the rectangular equation (x)^2 + (y - 22)^2 = (15)^2a) Write the parametric equations that correspond to this path and direction

(t, x, y) coordinates

x = 15sin((pi/40)t)y = -15cos((pi/40)t) + 22b) A bird is flying near the ferris wheel on a path described by x = .25t+10, y = 1.1t......Will the bird hit Izzy while he is on the ferris wheel? If yes, when and where?

x_1 = x_2

.25t+10 = 15sin((pi/40)t)

t = -100, -40, 20

y_1 = y_2

1.1t = -15cos((pi/40)t) + 22

t = 20, 28.112

x = .25(20) + 10 = 15

y = 1.1(20) = 22

YES at (15, 22) when t = 20 seconds## Minimum Distance

We are trying tograph the distance between two objects in relation to time, andinterpretthis graphto derive information. (For example,at t seconds position in time, how far apart are Object 1 and Object 2?)This example will correspond with the above "Simultaneous and Non-Simultaneous Solutions" problem (Example 2).

Given...Enter these equations into PARAMETRIC mode.

*x1 and y1 describe Object 1 and x2 and y2 describe Object 2.

In FUNCTION mode, enter the following equation into a blank y= space on your calculator (suggested y4 or y5)For TI-83/84 users:(You can find the variables for x1t, y1t, etc. under "vars," then "y-vars," then "parametric.")

For TI-89 users:adjust the window settings to model the situation!In this case use [xmin, xmax]= [-5,5] and [ymin, ymax]= [-4, 20].## Example 5

Locate and interpret the minimum over -3<t<0 and the minimum over 0<t<3.On the calculator, go to 2nd, Trace, Minimum.## Example 6

Approximate the time(s) when the objects are exactly 20 units apart.On the calculator, enter 20 into a y= function to graph the line y=20. Use the trace-minimum function as we did above to find where the intersections, because we know that when the two functions intersect the y value will be 20 to represent the distance between them, and we are looking for the x value aka time, for when this occurs.The intersections occur around.

**THINGS TO REMEMBER...Applying Minimum Distance in ContextThere are many real life situations where you can apply the context of minimum distance and graph the changing distance between two objects over time. Therefore there are theoretical intersections. For example, Izzy is trying to grab a fly and the graph depicts the distance between Izzy's hand and the fly over a time interval. If the graph crosses y=0 (distance is zero) twice, the second intersection does NOT truly exist because the fly must have already been grabbed the first time the distance between the hand and the fly is zero. It exists in theory and mathematically as it is a real point on the graph, but you are applying the graph to a real-life situation, make sure your answers are realistic!

Also, there are situations where in order for two objects to meet, the distance between them does NOT have to be zero. Imagine you are running and trying to reach out and grab your friend with your arm. The minimum distance graph is showing the distance between your body and your friend's. To make contact and grab her, your body does not have to be on tops of hers...because you reach out with your arm, allowing the distance for contact to be less than or equal to the radius that your arm provides. Although it is possible for you grab your friend at various distances such as y=1.5 ft or y=.5 ft, on a graph, the moment where you grab her will be the first time the distance is the length of your arm because you are answering this question IN CONTEXT. ^Refer to above example :)

## Projectile Motion

launched at an initial velocity (Vo)at aninitial angle to the ground (theta)itsposition at a time (t) can be modeled by the following:Gravity Constants

Gravity does not affect the horizontal componentof a projectile's motion. Therefore it'svelocityremainsconstant.Gravity affects the vertical componentof the projectile's motion. Therefore, we use the following constants in vertical motion equations:## Procedure for solving problems with projectile motion

1) Using the information given, create a diagram that visually represents the problem (it often helps)2) Write the x and y equations that describe the motion of the object

--modeled off of distance = rate x time

- y is the final vertical position
- y_0 is the initial vertical starting point

3) If the problem gives you the initial horizontal and vertical starting points, the initial velocity, and the the angle above the horizontal and asks you tofind the length of timethat the object is a projectile...- y = y_0 + (|v_0| sin(theta))t + (1/2)(g)(t)^2
- 0 = (given y value) + (given speed and angle)t + (1/2)(-32 ft/s/s OR -9.8 m/s/s)t^2
- Using the quadratic solver on your calculator, solve for t

4) If the problem asks you to find thehorizontal distancetraveled by the object- Since y = 0 when the object stops moving, you can use the time t that was calculated from step 3
- x = x_0 + (|v_0| cos(theta))t
- x_0 will almost always be 0
- x = 0 + (given speed and angle)(calculated value of t)

5) If the problem asks you if a ball in projectile motion will (for example) go over a fence, go into a basket, hit a pole, etc....Things to remember:Use correct units! (m/s with -9.8 m/s^2, ft/s with -32 ft/s^2)Make sure your calculator is in the correct angle mode (most often degrees with projectiles)## Example 7

Mr. Jain is up to bat. He hits a pitch that is waist high (1 meter above the ground) with an initial velocity of 36 m/s at an upward angle of 38 degrees. The ball travels to where the fence is 125m from home plate and 4m high. The fielder in the area can jump and catch a ball less that 3 m high. Write parametric equations that model the path of the ball then use a calculator to determine the result of the play (is it a home run, out, other?).1) Diagram:

2) Write x and y equations.

(hint: pay attention to parentheses because it can affect your answer)

3) (this problem is like step 5 listed above)

Time it takes to travel the horizontal distance of 125m. (this will help later)

125= (36 cos 38) t

t = 4.406 s

Now sub that time into the y equation

Y= (36 sin 38) (4.406) + 1 + .5 (-9.8) (4.406)^2

Y= 3.524 m

4) Find conditions: (use the details in the problem to figure this part out)

conditions for caught: when y is in interval [ 0, 3]

conditions for homerun: when y is greater than 4

conditions for "other": when y is in interval [3,4]

5) Therefore the result is "other" because 3.5 m is in between 3 and 4.

## Extra Resources

Additional Projectile Motion Exampleshttp://www.khanacademy.org/math/precalculus/v/parametric-equations-1

http://www.brightstorm.com/math/precalculus/vectors-and-parametric-equations/parametric-equations-and-motion-problem-3/

Graphical Significance of Projectile Motion

http://teachers.oregon.k12.wi.us/fishwild/UnitVIIDocs/TI89Projectile.pdf

Additional Parametric Motion Examples

http://www.youtube.com/watch?v=12b66OUOf7g

http://www.brightstorm.com/math/precalculus/vectors-and-parametric-equations/parametric-equations-and-motion/

## Rectilinear Motion

-acontinuous change of positionof a body so thatevery particleof the bodyfollows a straight-line path-all parts of the system

move at the same speed & in the same direction-also known as linear motion

-simple motion

Steps to graph

## Extra Resources

Rectilinear Motion Overviewhttp://www.answers.com/topic/rectilinear-motion

Rectilinear Motion Examples

http://www.youtube.com/watch?v=6ETHo8tCE3M

http://www.youtube.com/watch?v=7CBBOlTny4c

## Total distance

- the total distance the particle travels under the time restrictions## Total displacement

- the difference between the end position and beginning position## Example 8

x = 4, y = (t+5)(t-1)^2 defined for -5<t<51) t-x graph:

t-y graph:

2) -5

<t<5x: {4}

y: [0, 160]

3) x-y graph:

4) Motion of the object: Starts at (4, 0), moves upward between -5 < t < -3 to (4, 32), moves downward between -3 < t < 1 to (4, 0), then moves back upward between 1 < t < 5 to (4, 160).

5) Total distance = upward + downward + upward = 32 + 32 + 160 = 224 units

6) Total displacement = 160 - 0 = 160 units

## Extra Resources

Preview for next chapter:Applying Parametrics to form Conic Graphs

http://www.youtube.com/watch?v=57BiI_iD3-U

http://www.khanacademy.org/math/precalculus/v/parametric-equations-3

Visit Parametric Equations to form Conic Graphs:

http://pch-wiki.wikispaces.com/11.3+Content+-+Conics+B#Sketching%20the%20Graphs%20of%20Conics%20in%20Parametric%20Form:

For any extra parametric practice:http://www.sparknotes.com/math/precalc/parametricequationsandpolarcoordinates/problems.html

http://molinaro.net/Precalc_Parametric_Review.pdf

http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-02-834135-X&chapter=8&lesson=6&title=scq

http://www.mathplayground.com/ProjectTRIG/ProjectTRIGPreloader.html

Unit 10: Parametrics- Homework and Problem Set

https://sites.google.com/site/shsprecalchonors/unit-10-material

Primary authors of this page (as of 06/02/12):M. LiangA. HongJ. ChungM. ColleraR. HoyK. NoronhaR. Philipp