9.5 Parametric Equations

Transcription

1 Date: 9.5 Parametric Equations Syllabus Objective: 1.10 The student will solve problems using parametric equations. Parametric Curve: the set of all points xy,, where on an interval I (called the parameter interval) x f t and y g t are continuous functions of t Parameter: the variable t Parametric Equations: x f t and y g t Orientation: the directions that results from plotting the points as the values of t increase Graphing Parametric Equations Ex1: Graph x t, y t 1, 1 t Note: Choose appropriate values for t first. Then substitute in and find x and y. Make t x y a table: 0 1 Plot points: 1 Eliminating the Parameter 1. Solve one of the equations for t. (Or if a trig function, isolate the trig function.). Substitute for t in the other equation. (Or use an identity if a trig function.) Ex: Write the parametric equation as a function of y in terms of x. a) x t, y t 1 Solve for t in the x-equation (easier to solve for): x t t Substitute into the y-equation: Page 1 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

2 1 t b) x, y t t Solve for t in the x-equation (easier to solve for): Substitute into the y-equation: Ex3: Write the parametric equation as a function of y and graph. x tan, y tan 1, 0 Note: A parametric equation can be written in terms of θ instead of t. The x-equation is already solved for the trig function: x tan Substitute into the y-equation: y x 1 Graph: Using a Trig Identity Ex4: Eliminate the parameter. x 6cos t, y 6sin t, 0 t Solving for a trig function won t help, so we need to use the identity sin tcos t 1. Square both equations: Add the equations: Trig identity: Note: The graph is a circle. The parameter interval lets us know that it would go around 1 time. Writing a Parameterization Ex5: Find the parameterization of the line segment through the points A 1, and B, 3. Sketch a graph: Page of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

3 O A B P OP OA AP ; AP is a scalar multiple of AB, so OP OA t AB OP OA t AB x, y x, y 1 3 t, t These equations define the LINE. Find the parameter interval for the line segment: We want 1 x. x 1 3t: 1 1 3t t 0 1 3t t 1 So, 0t 1 Solution: Simulating Horizontal Motion Ex6: A dog is running on a horizontal path with the coordinates of his position (in meters) given by s 0.t 19t 100t 70 where 0 t 15. Use parametric equations and a graphing calculator to simulate the dog s motion. Choose any horizontal line to simulate the motion: We will choose y 3. Parametric Equations: x 0. t 19t 100t 70, y 3, 0 t 15 Graph (Calculator must be in Parametric mode): Note: To see the motion, change the type of line to a bubble. If you would like the bubble to move slower, make the Tstep smaller. Parametric Equations for Projectile Motion distance: x v t height: sin Note: On Earth, 0 cos g 3 ft/sec or g 9.8 m/sec. 1 y gt v t h Ex7: A golf ball is hit at 150 ft/sec at a 30 angle to the horizontal. a) When does it reach its maximum height? Height: sin y gt v0 t h0 ( h 0 0 because a golf ball is hit from the ground) Page 3 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

4 Simplify: b Maximum height is at vertex: t a b) How far does it go before it hits the ground? Hits the ground when y 0 : Note: We could have doubled the time it took for the ball to reach its highest point! Distance: x v 0 cos t c) Does the ball hit a 6 ft tall golfer, standing directly in the path of the ball 580 feet away? Find the time it takes for the ball to be 580 ft away: Find the height of the ball at this time: No the ball misses him by about feet. x v 0 cos t Application: Ferris Wheel Ex8: Ryan is on a Ferris wheel, alone, still looking for someone to share it with, of radius 0 ft that turns counterclockwise at a rate of one revolution every 4 sec. The lowest point of the Ferris wheel (6 o clock) is 10 ft above ground level at the point (0, 10) on a rectangular coordinate system. Find the parametric equations for the position of Ryan as a function of time t (in seconds) if the Ferris wheel starts (t = 0) with Ryan at the point (0, 30). Time to complete one revolution = 4 sec: When t 0, x 0 & y 30 : 4 1 You Try: A baseball is hit at 3 ft above the ground with an initial speed of 160 ft/sec at an angle of 17 with the horizontal. Will the ball clear a 0-ft wall that is 400 ft away? QOD: How would you write a parametrization for a semicircle? Reflection: Page 4 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

5 Date: 9.5 Parametric Equations Continued Syllabus Objectives: 1.10 The student will solve problems using parametric equations. 1.5 The student will find the inverse of a given function. 1.6 The student will compare the domain and range of a given function with those of its inverse. Parametric Equations: a pair of continuous functions that define the x and y coordinates of points in a plane in terms of a third variable, t, called the parameter. Ex1: Find xy, determined by the parameters t 3 for the function defined by the equations x t y t 1, 3. Find a direct relationship between x & y and indicate if the relation is a function. Then graph the curve. Create a table for t, x, & y. t xt 1 y t Solve for t and substitute to find a direct relationship between x and y. x t1 y t 3 y Graph by plotting the points in the table. Application Parametric Equations Ex: A stuntwoman drives a car off a 50 m cliff at 5 m/s. The path of the car is modeled by the equations x t y t 5 & How long does it take to hit the ground and how far from the base of the cliff is the impact? Page 5 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

6 The car hits the ground when y 0. The distance from the base of the cliff is x. Parametric Equations on the Graphing Calculator Ex3: Graph the situation in the previous example on the calculator. Change to Parametric Mode. Type each equation into the Y= menu and choose an appropriate window. This is the path of the car. Select the best viewing window for the graph. Reflection: Page 6 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

7 Date: 9.6 Polar Coordinates Syllabus Objectives: 3.3 The student will differentiate between polar and Cartesian (rectangular) coordinates. 6. The student will transform functions between Cartesian and polar form. 6.4 The student will solve real-world application problems using polar coordinates. Polar Coordinate: r, ; r: the directed distance from the pole (origin); θ: the directed angle from the polar axis (x-axis) Plotting Points on a Polar Graph Ex1: Plot the points 3, 3 A, B 8, 40, & C, 5 6. Point A: Start at the polar axis and go counter-clockwise 3 (70 ). Place the point 3 units from the pole (origin). Point B: Start at the polar axis and go clockwise 40. Place the point 8 units from the pole. (Note: Each radius drawn in the grid is 15.) Point C: Start by going counter-clockwise 5 6 (150 ) from the polar axis. Place a point units from the pole. Because r, you must place the point on the opposite side of the pole. Writing the Polar Coordinates of a Point Ex: Find four different polar coordinates of P. 6 4 P Page 7 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

8 Note: There are infinitely many correct answers! Polar Conversions Polar to Rectangular: Rectangular to Polar: rcos x rsin y tan y x 1 x y r r x y Converting from Polar to Rectangular Coordinates Ex3: Convert to rectangular coordinates. a) 5,45 rcos x rsin y b) 3, 3 rcos x rsin y Converting from Rectangular to Polar Coordinates (Note: Be careful with the quadrant!) Ex4: Convert to polar coordinates. tan y x 1 a) 1, 3 x y r r x y is in Quadrant II, so the polar coordinates are. b) 0, 4 This point is on the negative y-axis, so we know. Note: There are other possible answers to these! Page 8 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

9 Converting from Polar to Rectangular Equations Ex5: Convert the equations and sketch the graph. a) r r b) x rcos 4 y rsin c) r sec r 1 sec r cos 1 cos r Graphing in Polar Coordinates on the Calculator We will check our graphs above. Calculator must be in Polar mode. a) c) Note: We cannot check the graph of b) on the calculator, but the line y 45 for all values of r. 4 Converting from Rectangular to Polar Equations Ex6: Convert the equations. a) x 4 x rcos : r cos 4 x represents the angle Page 9 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

10 3x6y 0 b) 3x6y 0 c) x y 1 5 Expand: Substitute: So r 0 or r 4cos sin 0. But r 0 is a single point. So Application: Finding Distance Ex7: The location of two ships from the shore patrol station, given in polar coordinates, are mi, 150 & 3mi, 80. Find the distance between the ships. Sketch a diagram: Note: The angle between the ships (from the patrol station) is Using the Law of Cosines: You Try: Convert the coordinates. Polar:, ; Rectangular: 7,7 QOD: How could you write an expression for all of the possible polar coordinates of a point? Reflection: Page 10 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

11 Date: 9.7 Polar Graphs Syllabus Objectives: 6.3 The student will sketch the graph of a polar function and analyze it. 3 Main Graphs: Heart or loop (limacon), Roses (daisies), Fake Rose (lemniscates) r a bsin Equation: r a bcos a 1 Loop less than loop b a 1 Heart Great Heart b Ex.1: Transform to Cartesian r (4cos ) I. Heart or Loop Graphing Polar Curves Ex. Graph r 1 cos Ex. 3 Graph r 1 cos Page 11 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

12 Ex4: Graph and find the domain, range, symmetry, and maximum r-value. a) r 4cos Domain: Range: Max r-value: Symmetry: Substitute. Symmetric about the (This curve is called a limaçon.) II. Roses (daisies) r asin n r a cos n If n is odd. n= # of petals If n is even. n = # of petals Ex. 5a.) r sin3 Page 1 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

13 b) r 6sin 3 θ r Domain: Range: Max r-value: Symmetry: Substitute r &. Symmetric about This curve is called a rose. III. Fake Rose (lemniscates) r r a a sin cos Ex. 6 a.) r 4sin Page 13 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9

14 b) r 4cos Domain: Range: Max r-value: Symmetry: Substitute. Symmetric about Substitute r. r 4cos Symmetric about origin This curve is called a lemniscate. Substitute r &. r 4cos r 4cos Symmetric about y-axis Tests for Symmetry of Polar Curves 1. Symmetry about x-axis: r, is equivalent to r,. Symmetry about y-axis: r, is equivalent to 3. Symmetry about origin: r, is equivalent to r, r, Classifications of Polar Curves Limaçon Curves: r a bsin and r a bcos Rose Curves: r a cos n and r asin n Petals: odd = n and even = n Lemniscate Curves: r a cos and r a sin You Try: Use your graphing calculator to explore variations of r a sin n. Describe the effects of changing the window, the θ-step, a, n, and changing sin to cos. QOD: Are all polar curves bounded? Explain. Reflection: Page 14 of 14 Precalculus Graphical, Numerical, Algebraic: Larson Chapter 9