10.1+Content+-+Parametrics

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Add Parametrics content to this page. See specific details on the home page. media type="custom" key="18775180" media type="custom" key="18903860" = Parametrics (Unit 10) Learning Targets =
 * Quick Instructions**
 * To sketch the graphs of parametric equations.
 * To describe the motion modeled by parametric equations.
 * To convert equations from parametric form and x-y form.
 * To apply parametric equations to solve problems.
 * To use the calculator as a tool to analyze and interpret parametric equations.

= Parametric Equations = math x = f\left( t \right)\;\;{\text{and}}\;\;y = g\left( t \right) math
 * equations where the **x- and y- coordinates** of a point on a curve are given as **separate equations in terms of an independent variable** (called a parameter).

Graphs of parametric equations are often used to **model location and motion over time**:
 * The t-x graph tells you how values of t affect x-coordinate values
 * The t-y graph tells you how values of t affect y-coordinate values
 * The **x-y graph** tells you the **relationship between x and y** but does not describe the object's path
 * The purpose of the **parameter** t is **to tell you the effect of an independent variable** on x and y

Procedure for graphing parametric equations
When given a t-x equation and a t-y equation 1) 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/Hide

math x = \frac{4}{t-2},\,y = t+1 math

1) t-x graph: 

t-y graph:

2) Eliminate the parameter t and write the rectangular (x-y) form math \\ x = \frac{4}{t-2} \\ t = \frac{4}{x}+2 \\ \\ y = t+1 \\ \boxed{y = \frac{4}{x}+3} math

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). math \\ x \in \left(-\infty, 0\right) \cup \left(0, \infty\right) \\ y \in \left(-\infty, 3\right) \cup \left(3, \infty\right) math

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) = 1// //So x^2 = (4^2)cos^2(t)// //y^2 = (4^2)sin^2(t)//

//(x^2)/16 + (y^2)/16 = 1// //**x^2 + y^2 = 4^2**//

//3) t = all real #s// //x: [ -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...** math \\ \text{For example:} \\ x = 3 - t, \; y = 2 + t\\ \text{and} \\ x = t, \; y = 5 - t \\ \text{both produce an x-y equation of \boxed{y = 5 - x}} math
 * **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 changes**
 * **The 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 Unit **
 * []
 * []
 * [|http://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx]


 * Parametric Equation Examples **
 * []
 * []

Parametric vs Function Mode on a Calculator

 * In **function** mode, a **y-equation** is **defined in terms of x**
 * In **parametric** mode, for each parametric equation, there is **a x-equation and y-equation**. Both are **defined in terms of the parameter t**

Sequential vs. Simultaneous Method of Graphing on a Calculator

 * **Sequential** - Graphs **parametrics one at a time**
 * **Simultaneous** - Graphs **parametrics at the same time**
 * **TI-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

 * When graphing parametric equations, the **current location of the object is surrounded by a circle (beach ball)** and leaves behind a trail of the previous locations
 * Helps to illustrate the location of __simultaneous__ and __non-simultaneous solutions__ of a set of parametric equations
 * For both instructions below, first setup parametric mode. For help, look below.
 * **TI-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 = []

Extra Resources
Graphical 'Mode' Significance []

= Graphing Parametrics and Functions on the TI-89 = []

= Simultaneous Solutions and Non-Simultaneous Solutions =
 * __**Simultaneous solutions**__ occur at the **same place** at the **same time**
 * __**Non-simultaneous Solutions**__ occur at the **same place** but at **different times**

Procedure for finding simultaneous solutions

 * 1) Set X1=X2 and solve for the parameter t
 * 2) Set Y1=Y2 and solve for the parameter t
 * 3) If the parameter found from the x equations equals the parameter found from the y equations, there is a simultaneous solution at that value of t
 * 4) Once you know there is a simultaneous solution, insert the t you found into an x equation and a y equation (they must be from the same function e.g. x2 and y2) to find the coordinates of the collision at the time, t

Procedure for non-simultaneous solutions

 * 1) Eliminate the parameter so you are left with a system of two x-y equations
 * 2) Then by elimination, or linear combination, solve for one variable
 * 3) Substitute that value in an equation to solve for the other variable
 * 4) Answer will normally be a specific point or set of points

Example 3
Determine all simultaneous and non-simultaneous solutions for the set of parametric equations: math \\ X_1= t - 2 \\ Y_1= t^2 - 4 \\ \\ X_2 = t^2 - 8 \\ Y_2 = 12 - 3t^2 math Steps 1) and 2), set y and x equations equal to each other and solve for t math \\ X_1= X_2 \\ t - 2 = t^2 - 8 \\ 0 = t^2 - t - 6 \\ 0= (t - 3)(t + 2) \\ \boxed{t = -2,\,3} \\ \\ Y_1=Y_2 \\ t^2 - 4 = 12 - 3t^2 \\ 4t^2 = 16 \\ t^2 = 4 \\ \boxed{t= -2,\,2 \\} math
 * For simultaneous:**

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): math \\ X_1 = t - 2 \\ X_1 = (-2) - 2 \\ X_1 = -4 \\ \\ Y_1 = t^2 - 4 \\ Y_1 = (-2)^2 - 4 \\ Y_1 = 0\\ \\ \boxed{t = -2 \; at \; (-4, 0)} math

1) Eliminate parameters to have two y-x equations math \\ Y_1 = (X_1 + 2)^2 - 4 \\ Y_2 = 12 - 3 (X_2+ 8) \\ math
 * For non-simultaneous:**

2) Solve by combination or elimination (where Y1 = Y2 and X1 = X2) math \\ x^2 + 4x + 4 - 4 = 12 - 3x - 24 \\ x^2 + 4x = - 3X - 12 \\ x^2 + 7x + 12 = 0 \\ 0= (x + 3) (x + 4) \\ x = -3,\,-4 \\ math

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)^2 a) Write the parametric equations that correspond to this path and direction 

(t, x, y) coordinates
 * (0, 0, 7)
 * (20, 15, 22)
 * (40, 0, 37)
 * (60, -15, 22)


 * x = 15sin((pi/40)t)**
 * y = -15cos((pi/40)t) + 22**

b) 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 **

 =<range type="comment" id="873598">Minimum Distance = We are trying to **graph the distance between two objects in relation to time**, and **interpret** this graph **to derive information**. (For example, **at t seconds position in time, how far apart are Object 1 and Object 2?**)

math \\ \sqrt[]{\left(X_1 - X_2\right)^2\,+\,\left(Y_1-Y_2\right)^2} math

This example will correspond with the above "Simultaneous and Non-Simultaneous Solutions" problem (Example 2).

//Given...// math \\ X_1 = t-2 \\ Y_1 = t^2 - 4 \\ X_2 = t^2 - 8 \\ Y_2 = 12 - 3t^2 math 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)__

math \\ \sqrt{((x1t(x)-x2t(x))^2 + (y1t(x)-y2t(x))^2)} math (You can find the variables for x1t, y1t, etc. under "vars," then "y-vars," then "parametric.")
 * For TI-83/84 users:**

math \\
 * For TI-89 users: **

\sqrt{((xt1(x)-xt2(x))^2 + (yt1(x)-yt2(x))^2)} math
 * Make sure you **adjust the window settings to model the situation**! //In this case use [xmin, xmax]= [-5,5] and [ymin, ymax]= [-4, 20].//
 * Sketch the graph (Refer to attached picture to complete this problem.)
 * Label and interpret the axes; x represents time while y represents distance.

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.// math \\ \text{The first minimum is at } \boxed{(-2, 0)} \text{ This means that the distance between the two objects is 0 units and they intersect at this time.} \\ \text{The second minimum is at } \boxed{(2.04, 3.94)} \text{ This means that the distance between the two objects is 3.93 units and they intersect at this time.} math

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. math \boxed{t = -2 \text{ and } t = 3} math

__**THINGS TO REMEMBER...Applying Minimum Distance in Context__ There 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 =
 * For any projectile **launched at an initial velocity (Vo)** at an **initial angle to the ground (theta)** its **position at a time (t) can be modeled by the following**:

math \\ |V_0| \cos(\theta) = |V_x| \\ |V_0| \sin(\theta) = |V_y| \\ x = |V_0|\cos(\theta) t + x_0 \\ x = |V_x| t + x_0 \\ y = |V_0|\sin(\theta) t + y_0 + \frac{1}{2} g t^2 \\ y = |V_y| t + y_0 + \frac{1}{2} g t^2 math Gravity Constants math \\ g = -9.8 m/s/s \\ g = -32 ft/s/s math 
 * **Gravity does not affect the horizontal component** of a projectile's motion. Therefore it's **velocity** remains **constant**.
 * **Gravity affects the vertical component** of 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 math \boxed{x = x_0 + (|v_0|\cos(\theta))t} math math \boxed{y = y_0 + (|v_0|\sin(\theta))t + \frac{1}{2}(g)(t)^2} math 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 to **find the length of time** that the object is a projectile... 4) If the problem asks you to find the **horizontal distance** traveled by the object 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....
 * x is the final horizontal position
 * x_0 is the initial horizontal starting point
 * |v_0| is the magnitude of the initial velocity at which the object travels
 * t is the length of time the object travels
 * y is the final vertical position
 * y_0 is the initial vertical starting point
 * 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
 * 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)
 * Calculate the length of time that the object is in the air (as described in step 3) or how long it takes for the object to reach the fence, basket, pole, etc.
 * To do the latter...
 * The problem will most likely give you the distance between the moving object and the obstacle along with the initial velocity and angle
 * Set x equal to that distance
 * x = x_0 + (|v_0| cos(theta))t
 * (given x value) = 0 + (given speed and angle)t
 * Solve for t
 * y = y_0 + (|v_0| sin(theta))t + (1/2)(g)(t)^2
 * y = (given y value) + (given speed and angle)(calculated value of t) + (1/2)(-32 OR -9.8)(calculated value of t)^2 = #
 * Compare the # and the given distance to see if the object will reach that point at a certain time


 * 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)**



<span style="color: #800080; font-family: Arial,Helvetica,sans-serif;">Example 7
<span style="font-family: Arial,Helvetica,sans-serif;">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?). <span style="font-family: Arial,Helvetica,sans-serif;">1) Diagram: 2) Write x and y equations. math \\ x(t) = (36 \cos(38\textdegree)) t \\ y(t) = (36 \sin(38\textdegree)) t + 1 + \frac{1}{2} ( -9.8) t^2 math

<span style="font-family: Arial,Helvetica,sans-serif;">(hint: pay attention to parentheses because it can affect your answer)

<span style="font-family: Arial,Helvetica,sans-serif;">3) (this problem is like step 5 listed above) <span style="font-family: Arial,Helvetica,sans-serif;">Time it takes to travel the horizontal distance of 125m. (this will help later) <span style="font-family: Arial,Helvetica,sans-serif;">125= (36 cos 38) t <span style="font-family: Arial,Helvetica,sans-serif;">t = 4.406 s <span style="font-family: Arial,Helvetica,sans-serif;">Now sub that time into the y equation <span style="font-family: Arial,Helvetica,sans-serif;">Y= (36 sin 38) (4.406) + 1 + .5 (-9.8) (4.406)^2 <span style="font-family: Arial,Helvetica,sans-serif;">Y= 3.524 m

<span style="font-family: Arial,Helvetica,sans-serif;">4) Find conditions: (use the details in the problem to figure this part out) <span style="font-family: Arial,Helvetica,sans-serif;">conditions for caught: when y is in interval [ 0, 3] <span style="font-family: Arial,Helvetica,sans-serif;">conditions for homerun: when y is greater than 4 <span style="font-family: Arial,Helvetica,sans-serif;">conditions for "other": when y is in interval [3,4]

<span style="font-family: Arial,Helvetica,sans-serif;">5) Therefore the result is "other" because 3.5 m is in between 3 and 4.

Extra Resources
<span style="font-family: Arial,Helvetica,sans-serif;">Additional Projectile Motion Examples <span style="font-family: Arial,Helvetica,sans-serif;">[] <span style="font-family: Arial,Helvetica,sans-serif;">[] Graphical Significance of Projectile Motion [] <span style="font-family: Arial,Helvetica,sans-serif;">Additional Parametric Motion Examples <span style="font-family: Arial,Helvetica,sans-serif;">[] []

=<span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif;">Rectilinear Motion = -a **continuous change of position** of a body so that **every particle** of the body **follows 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 math \[y=at^{3}+bt^{2}+ct+d\] math
 * 1) you should have one equation like this: x=c (c is a constant)
 * 2) and another equation like this, which you should manually graph:


 * The graph will move along the path of x=c and the motion will model the motion of the y-t graph
 * The x-y graph will be a straight line, x=c, but the particle will change direction where the y-t graph turns



Extra Resources
<span style="font-family: Arial,Helvetica,sans-serif;">Rectilinear Motion Overview <span style="font-family: Arial,Helvetica,sans-serif;">[] <span style="font-family: Arial,Helvetica,sans-serif;">Rectilinear Motion Examples <span style="font-family: Arial,Helvetica,sans-serif;">[] <span style="font-family: Arial,Helvetica,sans-serif;">[]

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 __<__ 5**

1) t-x graph:

t-y graph:

2) -5 __<__ t __<__ 5 x: {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

= <range type="comment" id="898180">Extra Resources = Applying Parametrics to form Conic Graphs <span style="font-family: Arial,Helvetica,sans-serif;">[] <span style="font-family: Arial,Helvetica,sans-serif;">[] <span style="font-family: Arial,Helvetica,sans-serif;">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: [] [] [] [] Unit 10: Parametrics- Homework and Problem Set []
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 16px;">Preview for next chapter: **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 16px;">For any extra parametric practice: **


 * Primary authors of this page (as of 06/02/12):**
 * **M. Liang**
 * **A. Hong**
 * **J. Chung**
 * **M. Collera**
 * **R. Hoy**
 * **K. Noronha**
 * **R. Philipp**