7-5 Parametric Equations
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1 3. Sketch the curve given by each pair of parametric equations over the given interval. Make a table of values for 6 t 6. t x y Plot the (x, y) coordinates for each t value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as t moves from 6 to 6. esolutions Manual - Powered by Cognero Page 1
2 6. x = 2t 2 and y = 6; 6 t 6 Make a table of values for 6 t 6. t x y Plot the (x, y) coordinates for each t value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as t moves from 6 to 6. esolutions Manual - Powered by Cognero Page 2
3 Write each pair of parametric equations in rectangular form. Then graph the equation and state any restrictions on the domain. 9. x = 2t 5, y = t Solve for t in the parametric equation for x. Substitute for t in the parametric equation for y. Make a table of values to graph y. x y x y Plot the (x, y) coordinates and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as t increases. Since substituting larger values for t in x = 2t 5 produces increasing values of x, the orientation moves from left to right. esolutions Manual - Powered by Cognero Page 3
4 12. x = t 2 + 1, y = 4t + 3 Solve for t in the parametric equation for y. y = 4t + 3 y 3 = 4t = t Substitute for t in the parametric equation for x. Make a table of values to graph y. x y , , , 15 Plot the (x, y) coordinates and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as t increases. Since substituting larger values for t in y = 4t + 3 produces decreasing values of y, the orientation moves from top to bottom. 15. x = 4t 2, y = + 9 Solve for t in the parametric equation for y. esolutions Manual - Powered by Cognero Page 4
5 Substitute for t in the parametric equation for x. Make a table of values to graph y. x y , , , , 11.4 Plot the (x, y) coordinates and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as t increases. Since substituting larger values for t in y = + 9 produces increasing values of y, the orientation moves from bottom to top. esolutions Manual - Powered by Cognero Page 5
6 Write each pair of parametric equations in rectangular form. Then graph the equation. 18. x = 3 cos and y = 5 sin Solve the equations for cos θ and sin θ. Then use a trigonometric identity. x = 3 cos = cos y = 5 sin = sin sin 2 θ + cos 2 θ = 1 + = 1 + = 1 The equation is the general form of an ellipse that has a center at the origin, a vertical major axis of length 10, and a horizontal minor axis of length 6. Make a table of values for 0 θ 2π. θ x y π π 3 0 Plot the (x, y) coordinates for each θ value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as θ moves from 0 to 2π. esolutions Manual - Powered by Cognero Page 6
7 21. x = 3 cos and y = 3 sin Solve the equations for cos θ and sin θ. Then use a trigonometric identity. x = 3 cos = cos y = 3 sin = sin sin 2 θ + cos 2 θ = 1 + = 1 + = 1 The equation can be written in the general form of a circle, x 2 + y 2 = 9, that has a center at the origin and radius 3. Make a table of values for 0 θ 2π. θ x y π π 3 0 Plot the (x, y) coordinates for each θ value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as θ moves from 0 to 2π. esolutions Manual - Powered by Cognero Page 7
8 24. x = 10 sin and y = 9 cos Solve the equations for cos θ and sin θ. Then use a trigonometric identity. y = 9 cos = cos x = 10 sin = sin sin 2 θ + cos 2 θ = 1 + = 1 + = 1 The equation is the general form of an ellipse that has a center at the origin, a horizontal major axis of length 20 and a vertical minor axis of length 18. Make a table of values for 0 θ 2π. θ x y π π 0 9 Plot the (x, y) coordinates for each θ value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve as θ moves from 0 to 2π. esolutions Manual - Powered by Cognero Page 8
9 Use each parameter to write the parametric equations that can represent each equation. Then graph the equations, indicating the speed and orientation. 27. t = 8x; y 2 = 9 x 2 t = 8x = x y 2 = 9 x 2 y 2 = 9 y 2 = 9 y = Make a table of values for t. t x y Plot the (x, y) coordinates for each t value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve. 30. t = 4x + 7; y = t = 4x + 7 t 7 = 4x = x y = = esolutions Manual - Powered by Cognero Page 9
10 y = y = y = y = y = y = y = + y = + Make a table of values for t. t x y Plot the (x, y) coordinates for each t value and connect the points to form a smooth curve. The arrows in the graph indicate the orientation of the curve. esolutions Manual - Powered by Cognero Page 10
11 33. FOOTBALL Delmar attempts a 43 yard field goal. He kicks the ball at a angle with an initial speed of 70 feet per second. The goal post is 15 feet high. Is the kick long enough to make the field goal? To determine whether Delmar makes the field goal, you need the horizontal distance that the ball has traveled when the height of the ball is 15 feet. First, write a parametric equation for the vertical position of the ball. y = tv 0 sin θ gt 2 + h 0 = t(70) sin 41 (32)t Graph the equation for the vertical position and the line y = 15. The curve will intersect the line in two places. The second intersection represents the ball as it is moving down toward goal post. Use 5: intersect function on the CALC menu to find the second point of intersection with y = 15. The value is about 2.49 seconds. Determine the horizontal position of the ball at 2.49 seconds. x = tv 0 cos θ = 2.49(70) cos The ball will travel a distance of about 132 feet before the height reaches 15 feet a second time. There are 3 feet in a yard, so the distance of the field goal is 43 3 or 129 feet. Since the vertical position of the ball will not reach 15 feet until it achieves a horizontal distance of 132 feet, Delmar will make the field goal. esolutions Manual - Powered by Cognero Page 11
12 Write each pair of parametric equations in rectangular form. Then state the restriction on the domain. 36. x = y = 3t 8 Solve for t. x = x 2 = t 7 x = t Substitute. y = 3t 8 y = 3(x 2 + 7) 8 y = 3x y = 3x 2 29 From the parametric equation x =, the only possible values for x are values greater than or equal to zero. The domain of the rectangular equation needs to be restricted to x x = y = 2t 4 Solve for t. x = = log (t + 2) = t Substitute. y = 2t 4 = t y = 2 4 y = y = 2 8 From the parametric equation x = equation needs to be restricted to x 0., x can be all real numbers except 0. The domain of the rectangular esolutions Manual - Powered by Cognero Page 12
13 Write a set of parametric equations for the line or line segment with the given characteristics. 42. line with a slope of 0.5 that passes through (3, 2) Make a table of values. Let t = 0 represent the first point (3, 2) and let t be in intervals of 1. t x y Since the slope of the line is given as 0.5, the y values will decrease by 0.5 unit each time x increases by 1 unit. Find an equation for x in terms of t. Since x increases by increments of 1, m = 1. x = mt + b 3 = (1)(0) + b 3 = b An equation for x is x = t + 3. Repeat the process to find an equation for y in terms of t. Since y decreases by increments of 0.5, m = 0.5. y = mt + b 2 = ( 0.5)(0) + b 2 = b An equation for y is y = 0.5t 2. esolutions Manual - Powered by Cognero Page 13
14 Match each set of parametric equations with its graph. 45. x = cos 2t, y = sin 4t Use a graphing calculator to graph the parametric equations x = cos 2t and y = sin 4t. Graph in radians. The answer choice that resembles the graph is b. 48. x = cos 4t, y = sin 3t Use a graphing calculator to graph the parametric equations x = cos 4t and y = sin 3t. Graph in radians. The answer choice that resembles the graph is c. 51. SOCCER The graph below models the path of a soccer ball kicked by one player and then headed back by another player. The path of the initial kick is shown in blue, and the path of the headed ball is shown in red. a. If the ball is initially kicked at an angle of, find the initial speed of the ball. b. At what time does the ball reach the second player if the second player is standing about 17.5 feet away? c. If the second player heads the ball at an angle of, an initial speed of 8 feet per second, and at a height of 4.75 feet, approximately how long does the ball stay in the air from the time it is first kicked until it lands? esolutions Manual - Powered by Cognero Page 14
15 a. The horizontal position of the ball can be modeled by x = tv 0 cos θ. Since the point (7, 5.77) is provided at t = 0.4, solve for v 0 using x = 7, t = 0.4, and θ =. x = tv 0 cos θ 7 = 0.4 v 0 cos 50 = v = v 0 The initial velocity of the ball is 27.2 feet per second. b. To find the time the ball reaches the second player, solve for t using the horizontal position function. x = tv 0 cos θ 17.5 = t(27.2) cos 50 = v = v 0 The ball will reach the second player in about 1 second. c. To find the time that the ball is in the air after the second player hits it, solve for t when the vertical position function equals 0. Since velocity is given in feet, use 32 feet per second squared for g. y = tv 0 sin θ gt 2 + h 0 0 = t(8) sin 75 (32)t = 7.73t 16t Graph the equation for the vertical position. Use 2: zero function on the CALC menu to find the time t for when the ball will hit the ground. The value is about 0.84 second. The ball was in the air for 1 second before it was hit by the second player, and then stayed in the air for an additional 0.84 second. In total, the ball was in the air for about 1.84 seconds. esolutions Manual - Powered by Cognero Page 15
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