Concept Category 4 (textbook ch. 8) Parametric Equations

Parametric Equations Write parametric equations. Graph parametric equations. Determine an equivalent rectangular equation for parametric equations. Determine parametric equations for a rectangular equation. Determine the location of a moving object at a specific time. 2

It is the bottom of the ninth inning, with two outs and two men on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately 45 degrees to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. 3

Algebra is really about relationships. How are things connected? Do they move together, or apart, or maybe they re completely independent? Normal equations assume an input to output connection. That is, we take an input (x=3), plug it into the relationship (y=x 2 ), and observe the result (y=9). But is that the only way to see a scenario? The setup y=x 2 implies that y only moves because of x. But it could be that y just coincidentally equals x 2, and some hidden factor is changing them both (the factor changes x to 3 while also changing y to 9). 4

As a real world example: For every degree above 70, our convenience store sells x bottles of sunscreen and x 2 pints of ice cream. We could write the algebra relationship like this: 2 ice cream sunscreen And it s correct but misleading! The equation implies sunscreen directly changes the demand for ice cream, when it s the hidden variable (temperature) that changed them both! 5

It s much better to write two separate equations sunscreen temperature 70 ice cream (t emperature 70) that directly point out the causality. The ideas temperature impacts ice cream and temperature impacts sunscreen clarify the situation, and we lose information by trying to factor away the common temperature portion. Parametric equations get us closer to the realworld relationship. 2 https://betterexplained.com/articles/a-quick-intuition-for-parametric-equations/ 6

Definition 7

Let s start with an easy one. Then create a rectangular equation 9

Solution: 11

Ex2) Graph the curve represented by the equations x 1 2 t, y t 2 3; 3 t 3. The rectangular equation is? 13

Ex2) Graph the curve represented by the equations x 1 2 t, y t 2 3; 3 t 3. The rectangular equation is y 4x 2 3, 3 2 x 3 2. 82

Example: Sketch the curve given by x = t + 2 and y = t 2, 3 t 3. t 3 2 1 0 1 2 3 x 1 0 1 2 3 4 5 y 9 4 1 0 1 4 9 y orientation of the curve 8 4-4 4 x Copyright by Houghton Mifflin Company, Inc. All rights reserved. 16

Eliminating the parameter is a process for finding the rectangular equation (in x and y) of a curve represented by parametric equations. x = t + 2 y = t 2 t = x 2 Parametric equations Solve for t in one equation. y = (x 2) 2 Substitute into the second equation. y = (x 2) 2 Equation of a parabola with the vertex at (2, 0) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 19

For your For information: your information pg. 2

Link: graphing conics for TIs https://education.ti.com/en- AU/software/details/en/6D27F3EB7E354D8D BDA5543AF744D0E3/83conicgraphing 22

PARAMETRIC CURVES for Trigonometry What curve is represented by the following parametric equations? x = cos t y = sin t 0 t 2π

t (angles) x = cos t y = sin t 0 also 360 45 90 135 180 225 270 315 24

If we plot points, it appears the curve is a circle. We can confirm this by eliminating t.

Notice that, in this example, the parameter t can be interpreted as the angle as shown.

From CC2, Trig Identity: cos 2 t + sin 2 t = 1 thus x 2 + y 2 = 1 This is the rectangular equation Thus, the point (x, y) moves on the unit circle x 2 + y 2 = 1

Trigonometric Parametric xt ( ) 4 cos t y(t) 8sin t 0 t 2 a) Sketch b) Find the rectangular equation 29

Let s just use the simple input values t x(t) y(t) 0 90 180 270 360 30

Trigonometric Parametric xt ( ) cos t 4 y(t) sin t 8 0 t 2 a) Sketch b) Find the rectangular equation 33

Trigonometric Parametric xt ( ) 4 cos t 4 y(t) 8sin t 8 0 t 2 a) Sketch b) Find the rectangular equation 36

Due 02/20th CC3 Model Cards Pick 2 problems Traveling at a speed of 10 knots, a ship proceeds south from its port for 2.5 hours and then changes course to true bearing of 130 degrees for 2 hours. At this time, how far from the port is the ship? *FYI: 1 knot = 1 nautical miles per hr = 1.15 miles per hr A plane is flying at a speed of 805.4 miles per hour on a true bearing of 204 degrees. If the plane s groundspeed is 908.8 miles per hour and its true bearing is 228 degrees, what are the speed and direction of the wind? The course for a boat race starts at point A and proceeds in the direction with true bearing of 152.5 degrees to point B, then in the direction with true bearing of 44.8 degrees to point C, and finally back to point A. Point C lies 128.5 kilometers directly east of point A. Find the total distance of the race course. 39

CC3 Model Cards Traveling at a speed of 10 knots, a ship proceeds south from its port for 2.5 hours and then changes course to true bearing of 130 degrees for 2 hours. At this time, how far from the port is the ship? distance: approx 40.84 knots; direction: approx -67.97 degrees Q2 = 292.03 degrees A plane is flying at a speed of 805.4 miles per hour on a true bearing of 204 degrees. If the plane s groundspeed is 908.8 miles per hour and its true bearing is 228 degrees, what are the speed and direction of the wind? Distance: approx 370.47 m/h; direction: approx -20.16 degrees Q2 = 159.84 degrees The course for a boat race starts at point A and proceeds in the direction with true bearing of 152.5 degrees to point B, then in the direction with true bearing of 44.8 degrees to point C, and finally back to point A. Point C lies 128.5 kilometers directly east of point A. Find the total distance of the race course. a is 119.64 km; c is 95.71km; total = 343.85km 40

2/14 th Practice for CC4 Graph the curve represented by the equations, then find the rectangular equation for each: a] x 5cos t y 2 sin t 0 t 2 b x t t y t t 2 ] 2 1 2 1 c t t 2 ] x(t) sin t y(t) 2 cos 0 2 d x t y t t t in 2 2 4 ] 1 2 [0,1] e] x cos t 2 y sin t 3 0 t 2 f ] x 3cos t y 2 sin t t in [0, 2 ] 41

B] solution for converting it to rectangular equation It is not possible to solve for x in this case, so we have to rewrite y y y 2t 1 1 2t y 1 2 t x 2 y 1 y 1 2 2 y y 1 y 1 4 2 4 2 2 1 3 4 4 2 2 y y Copyright by Houghton Mifflin Company, Inc. All rights reserved. 43

Happy Friday Monday Tuesday Ex) x(t) = 2sec t, y(t) = 4tan t *Sketch (it is not a circle or ellipse) 0 t 2 *Create a rectangular equation using the second trig identity: (sec t) 2 (tan t) 2 1 46

0 45 t x(t) = 2sec t y(t) = 4tan t 90 135 180 225 no solution so don t bother 270 no solution so don t bother 315 Hope no one forgets this : sec t 1 cos t 47

2 2 (sec t) (tan t) 1 x y sec t tan t 2 4 2 2 (x/ 2) (y/ 4) 1 48

Ex) x(t) = 2cot t, y(t) = 4csc t *Sketch (it is not a circle or ellipse) *Create a rectangular equation using the third trig identity: (csc t) 2 (cot t) 2 1 50

0 45 90 135 180 225 270 315 t x(t) = 2cot t y(t) = 4csc t cot t 1 tan t Just in case: 1 csc t sin t 51

2 2 (csc t) (cot t) 1 x y cot t csc t 2 4 2 2 (y/ 4) (x/ 2) 1 which is : 2 2 (x/ 2) (y/ 4) 1 52

If an object is dropped, thrown, launched etc. at a certain angle and has gravity acting upon it, the equations for its position at time t can be written as: v cos t y gt v sin t h o x o horizontal position 1 2 2 initial velocity angle measured from horizontal time vertical position gravitational constant which is 9.8 m/s 2 or 32 ft/s 2 initial height

Finding the parametric equations to describe the motion of a baseball A baseball is hit with an initial velocity of 140 ft/sec at an angle of 45 degrees to the horizontal, making contact 3 ft above the ground: 57

4. Cannot confirm that the hit was home run without knowing the size of the outfield. For simplicity s sake, we will use 400 ft as the distance between the outfield wall and home plate, and 10 feet for the height of the wall: set x = 400 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 60

Example 2 61

Example3) 63

By solving x, you will realize the ball will reach the fence in less than 3 seconds. So it will not clear the wall. 64

example 4) The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. A baseball is hit at a point 3 feet above the ground and leaves the bat at a speed of 150 feet per second at an angle of 15. The parametric equations for its path are? Graph the path of the baseball. Is the hit a home run? 25 20 15 10 y 5 0 50 100 150 200 250 300 350 400 x Copyright by Houghton Mifflin Company, Inc. All rights reserved. 65

Application: The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. A baseball is hit at a point 3 feet above the ground and leaves the bat at a speed of 150 feet per second at an angle of 15. The parametric equations for its path are x = 145t and y = 3 + 39t 16t 2. Graph the path of the baseball. Is the hit a home run? y (0, 3) 25 20 15 10 5 0 50 100 150 200 250 300 350 400 The ball only traveled 364 feet and was not a home run. (364, 0) x Home Run Copyright by Houghton Mifflin Company, Inc. All rights reserved. 66

Practice Now 2/22 Routine: For each equation, create a rectangular equation, and sketch t,x,y chart optional 2 1] x t 1, y t 4 t 4 2 2]x t, y t 1 4 t 4 3] x 4 2 cos t, y 2 3sin t 0 t 2 4] x 5sec t, y 6 tan t 0 t 2 5] x 4 csc t, y 3cot t 0 t 2 67

Application Practice 68

3] Rectangular equation method 2 : x 2 t x y t 1 x 1 t 70

More Practice 2/23 a] x 5cos t 3 y 2 sin t 4 0 t 2 b x t y t t 2 ] 2 3 3 3 c] x 2sec t 3 y 4 tan t 5 0 t 2 d] x 2 cot t 3 y 4 csc t 5 0 t 2 e x t y t t 2 ] 4 5 3 3 F] Lisa hits a golf ball off the ground with a velocity of 50 ft./sec at an angle of 35 from the horizon. (a) Write a set of parametric equations to model this situation. (b) Find when and where the ball will hit the ground (c) Find the max. height of the ball. When does this occur? 73

B] D] E] 74

Quiz yourself (no notes) Sketch the parametric equations: Convert the parametric equations to rectangular equations: 77

CC4: Conics Circles Parabolas Ellipse Hyperbolas 79

Circles The set of all points that are the same distance from the center. Standard Equation: 2 2 2 ( x h) ( y k) r CENTER: (h, k) RADIUS: r (square root) (h, k) r

Ellipses Salami is often cut obliquely to obtain elliptical slices, which are larger.

V a Center: (h,k) Ellipses Basically, an ellipse is a squished circle Standard Equation: (h, k) b CV (x - h) a 2 2 + c 2 = a 2 - b 2 (y - k) b 2 2 2 (cos t) (sin t) 1 2 =1 Formula for the Foci 2 2 x h y k a b a: major radius: the longer radius; the vertices are located at the ends of the major radii b: minor radius: the shorter radius; the co-vertices are located their ends 1

Quick Check Error Analysis #1, 2, 3: Graph = 1 point Mastery Check next Thursday Rectangular eq. = 1 point If #1 to 3 are all correct: #4a correct = 3- a + b = 3+ all correct = 4+ 83

History Early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve. In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.

Science On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

Hyperbolas 2 2 (sec t) (tan t) 1 2 2 (csc t) (cot t) 1 Looks like: two parabolas, back to back. Center: (h, k) Standard Equations: ( x h) a 2 2 2 ( y k) 2 b 1 ( y k) a 2 2 2 ( x h) 2 b 1 Opens LEFT and RIGHT (h, k) Opens UP and DOWN (h, k)

Hyperbolas Transverse Axis

There is a pattern 88

Sketch the Conics (Rectangular Eq.) c] x 2sec t 3 y 4 tan t 5 0 t 2 d] x 2cot t 3 y 4csc t 5 0 t 2 90

a x t y t Happy Friday Create a rectangular equation for each, then sketch: (label key features for each graph) 2 ] 3 1 b] x 3sec t 2 y 2 tan t 3 0 t 2 c] x 4 cos t 5 y 6sin t 2 0 t 2 d] x 7 cot t 1 y 3csc t 2 0 t 2 92

Create a rectangular equation for each graph, then create parametric equations 93

2 2 (sec t) (tan t) 1 x 2 y 2 sec t tan t 3 4 3sec t x 2 4 tan t y 2 2 2 (cos t) (sin t) 1 x 1 y cos t sin t 7 3 7 cos t x 1 7 cos t 1 x 3sin t 3sec t 2 x 4 tan t 2 y 2 2 (csc t) (cot t) 1 You might have a different solution for this problem y 2 x 1 csc t cot t 2 4 2 csc t y 2 4 cot t x 1 2 csc t 2 y 4 cot t 1 x y 96

Sketch the equations: CC Final Study Guide x 1 1] y 2(3) 4 CC1] y 2 Log ( x 1) 4 o CC 2] y 2 sin( x 90 ) 4 o CC 2] y 2 cos( x 90 ) 4 CC y x 2 4] 2( 1) 4 CC x y 2 4] 2( 1) 4 3 2 2 x 1 y 4 CC4] 1 2 2 2 2 x 1 y 4 CC4] 1 2 2 Know all these graphs and their key features for the finals 98

Horizontal asymptote Vertical Asymptote 99

Asymptotes 100

Mastery Check Thursday Mar. 8th Projectile Motion Kevin hits a baseball at 3 ft above the ground with an initial speed of 150ft/sec at an angle of 18 degrees with the horizontal: a) Write the parametric equations that represent the motion of the baseball b) What is the maximum height of the baseball? What is the time when it reaches its maximum height? How far is this from Kevin in the horizontal direction? c) When will it hit the ground? How far is this from Kevin in the horizontal direction? d) Will it clear a 20 ft wall that is 400 ft away? Create a sketch for this scenario. e) Create a rectangular equation 101

[CC1] Create an exponential or log equation for each graph [CC4] Create a rectangular, then parametric equations: 102

Mastery Check Thursday Mar. 8th Projectile Motion Answer Key Kevin hits a baseball at 3 ft above the ground with an initial speed of 150ft/sec at an angle of 18 degrees with the horizontal: a x t y t t o 2 o ] (150 cos18 ) 16 (150 sin18 ) 3 b] max height 36.53ft; t 1.45sec; about 206 ft away c]t 2.9sec; about 220 ft away o d ] No,if you solve t 400 (150cos18 ) t, then find y, y 7.35 ft only 2 x o x e] 16 (150 sin18 ) o 3 150 cos18 150 cos18 o 103