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Algebra and Trigonometry

10.7 Parametric Equations: Graphs

Algebra and Trigonometry10.7 Parametric Equations: Graphs
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Learning Objectives

In this section you will:
  • Graph plane curves described by parametric equations by plotting points.
  • Graph parametric equations.

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° 45° 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. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

Photo of a baseball batter swinging.
Figure 1 Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)

Graphing Parametric Equations by Plotting Points

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

How To

Given a pair of parametric equations, sketch a graph by plotting points.

  1. Construct a table with three columns: t,x(t),andy(t). t,x(t),andy(t).
  2. Evaluate x x and y y for values of t t over the interval for which the functions are defined.
  3. Plot the resulting pairs ( x,y ). ( x,y ).

Example 1

Sketching the Graph of a Pair of Parametric Equations by Plotting Points

Sketch the graph of the parametric equations x(t)= t 2 +1,y(t)=2+t. x(t)= t 2 +1,y(t)=2+t.

Analysis

As values for t t progress in a positive direction from 0 to 5, the plotted points trace out the top half of the parabola. As values of t t become negative, they trace out the lower half of the parabola. There are no restrictions on the domain. The arrows indicate direction according to increasing values of t. t. The graph does not represent a function, as it will fail the vertical line test. The graph is drawn in two parts: the positive values for t, t, and the negative values for t. t.

Try It #1

Sketch the graph of the parametric equations x= t ,y=2t+3,0t3. x= t ,y=2t+3,0t3.

Example 2

Sketching the Graph of Trigonometric Parametric Equations

Construct a table of values for the given parametric equations and sketch the graph:

x=2cost y=4sint x=2cost y=4sint

Analysis

We have seen that parametric equations can be graphed by plotting points. However, a graphing calculator will save some time and reveal nuances in a graph that may be too tedious to discover using only hand calculations.

Make sure to change the mode on the calculator to parametric (PAR). To confirm, the Y= Y= window should show

X 1T = Y 1T = X 1T = Y 1T =

instead of Y 1 =. Y 1 =.

Try It #2

Graph the parametric equations: x=5cost,y=3sint. x=5cost,y=3sint.

Example 3

Graphing Parametric Equations and Rectangular Form Together

Graph the parametric equations x=5cost x=5cost and y=2sint. y=2sint. First, construct the graph using data points generated from the parametric form. Then graph the rectangular form of the equation. Compare the two graphs.

Analysis

In Figure 5, the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.

Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.
Figure 5

Example 4

Graphing Parametric Equations and Rectangular Equations on the Coordinate System

Graph the parametric equations x=t+1 x=t+1 and y= t ,t0, y= t ,t0, and the rectangular equivalent y= x1 y= x1 on the same coordinate system.

Analysis

With the domain on t t restricted, we only plot positive values of t. t. The parametric data is graphed in blue and the graph of the rectangular equation is dashed in red. Once again, we see that the two forms overlap.

Try It #3

Sketch the graph of the parametric equations x=2cosθandy=4sinθ, x=2cosθandy=4sinθ, along with the rectangular equation on the same grid.

Applications of Parametric Equations

Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in x and y give an overall picture of an object's path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of x and y change depending on t, as the location of a moving object at a particular time.

A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of θ θ to the horizontal, with an initial speed of v 0 , v 0 , and at a height h h above the horizontal.

The path of an object propelled at an inclination of θ θ to the horizontal, with initial speed v 0 , v 0 , and at a height h h above the horizontal, is given by

x=( v 0 cosθ)t   y= 1 2 g t 2 +( v 0 sinθ)t+h x=( v 0 cosθ)t   y= 1 2 g t 2 +( v 0 sinθ)t+h

where g g accounts for the effects of gravity and h h is the initial height of the object. Depending on the units involved in the problem, use g=32ft/ s 2 g=32ft/ s 2 or g=9.8m/ s 2 . g=9.8m/ s 2 . The equation for x x gives horizontal distance, and the equation for y y gives the vertical distance.

How To

Given a projectile motion problem, use parametric equations to solve.

  1. The horizontal distance is given by x=( v 0 cosθ )t. x=( v 0 cosθ )t. Substitute the initial speed of the object for v 0 . v 0 .
  2. The expression cosθ cosθ indicates the angle at which the object is propelled. Substitute that angle in degrees for cosθ. cosθ.
  3. The vertical distance is given by the formula y= 1 2 g t 2 +( v 0 sinθ )t+h. y= 1 2 g t 2 +( v 0 sinθ )t+h. The term 1 2 g t 2 1 2 g t 2 represents the effect of gravity. Depending on units involved, use g=32 ft/s 2 g=32 ft/s 2 or g=9.8 m/s 2 . g=9.8 m/s 2 . Again, substitute the initial speed for v 0 , v 0 , and the height at which the object was propelled for h. h.
  4. Proceed by calculating each term to solve for t. t.

Example 5

Finding the Parametric Equations to Describe the Motion of a Baseball

Solve the problem presented at the beginning of this section. Does the batter hit the game-winning home run? Assume that the ball is hit with an initial velocity of 140 feet per second at an angle of 45° 45° to the horizontal, making contact 3 feet above the ground.

  1. Find the parametric equations to model the path of the baseball.
  2. Where is the ball after 2 seconds?
  3. How long is the ball in the air?
  4. Is it a home run?

Media

Access the following online resource for additional instruction and practice with graphs of parametric equations.

10.7 Section Exercises

Verbal

1.

What are two methods used to graph parametric equations?

2.

What is one difference in point-plotting parametric equations compared to Cartesian equations?

3.

Why are some graphs drawn with arrows?

4.

Name a few common types of graphs of parametric equations.

5.

Why are parametric graphs important in understanding projectile motion?

Graphical

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.

6.

{ x( t )=t y( t )= t 2 1 { x( t )=t y( t )= t 2 1

t t x x y y
3 3
2 2
1 1
0 0
1 1
2 2
3 3
7.

{ x( t )=t1 y( t )= t 2 { x( t )=t1 y( t )= t 2

t t 3 3 2 2 1 1 0 0 1 1 2 2
x x
y y
8.

{ x( t )=2+t y( t )=32t { x( t )=2+t y( t )=32t

t t 2 2 1 1 0 0 1 1 2 2 3 3
x x
y y
9.

{ x( t )=22t y( t )=3+t { x( t )=22t y( t )=3+t

t t 3 3 2 2 1 1 0 0 1 1
x x
y y
10.

{ x( t )= t 3 y( t )=t+2 { x( t )= t 3 y( t )=t+2

t t 2 2 1 1 0 0 1 1 2 2
x x
y y
11.

{ x( t )= t 2 y( t )=t+3 { x( t )= t 2 y( t )=t+3

t t 2 2 1 1 0 0 1 1 2 2
x x
y y

For the following exercises, sketch the curve and include the orientation.

12.

{ x(t)=t y(t)= t { x(t)=t y(t)= t

13.

{ x(t)= t y(t)=t { x(t)= t y(t)=t

14.

{ x(t)=5| t | y(t)=t+2 { x(t)=5| t | y(t)=t+2

15.

{ x(t)=t+2 y(t)=5| t | { x(t)=t+2 y(t)=5| t |

16.

{ x(t)=4sint y(t)=2cost { x(t)=4sint y(t)=2cost

17.

{ x(t)=2sint y(t)=4cost { x(t)=2sint y(t)=4cost

18.

{ x(t)=3 cos 2 t y(t)=−3sint { x(t)=3 cos 2 t y(t)=−3sint

19.

{ x(t)=3 cos 2 t y(t)=−3 sin 2 t { x(t)=3 cos 2 t y(t)=−3 sin 2 t

20.

{ x(t)=sect y(t)=tant { x(t)=sect y(t)=tant

21.

{ x(t)=sect y(t)= tan 2 t { x(t)=sect y(t)= tan 2 t

22.

{ x(t)= 1 e 2t y(t)= e t { x(t)= 1 e 2t y(t)= e t

For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.

23.

{ x( t )=t1 y( t )= t 2 { x( t )=t1 y( t )= t 2

24.

{ x( t )= t 3 y( t )=t+3 { x( t )= t 3 y( t )=t+3

25.

{ x(t)=2cost y(t)=sint { x(t)=2cost y(t)=sint

26.

{ x(t)=7cost y(t)=7sint { x(t)=7cost y(t)=7sint

27.

{ x(t)= e 2t y(t)= e t { x(t)= e 2t y(t)= e t

For the following exercises, graph the equation and include the orientation.

28.

x= t 2 ,y=3t,0t5 x= t 2 ,y=3t,0t5

29.

x=2t,y= t 2 ,5t5 x=2t,y= t 2 ,5t5

30.

x=t,y= 25 t 2 ,0<t5 x=t,y= 25 t 2 ,0<t5

31.

x(t)=t,y(t)= t ,t0 x(t)=t,y(t)= t ,t0

32.

x=2cost,y=6sint,0tπ x=2cost,y=6sint,0tπ

33.

x=sect,y=tant, π 2 <t< π 2 x=sect,y=tant, π 2 <t< π 2

For the following exercises, use the parametric equations for integers a and b:

x(t)=acos((a+b)t) y(t)=acos((ab)t) x(t)=acos((a+b)t) y(t)=acos((ab)t)
34.

Graph on the domain [ π,0 ], [ π,0 ], where a=2 a=2 and b=1, b=1, and include the orientation.

35.

Graph on the domain [ π,0 ], [ π,0 ], where a=3 a=3 and b=2 b=2 , and include the orientation.

36.

Graph on the domain [ π,0 ], [ π,0 ], where a=4 a=4 and b=3 b=3 , and include the orientation.

37.

Graph on the domain [ π,0 ], [ π,0 ], where a=5 a=5 and b=4 b=4 , and include the orientation.

38.

If a a is 1 more than b, b, describe the effect the values of a a and b b have on the graph of the parametric equations.

39.

Describe the graph if a=100 a=100 and b=99. b=99.

40.

What happens if b b is 1 more than a? a? Describe the graph.

41.

If the parametric equations x(t)= t 2 x(t)= t 2 and y( t )=63t y( t )=63t have the graph of a horizontal parabola opening to the right, what would change the direction of the curve?

For the following exercises, describe the graph of the set of parametric equations.

42.

x(t)= t 2 x(t)= t 2 and y( t ) y( t ) is linear

43.

y(t)= t 2 y(t)= t 2 and x( t ) x( t ) is linear

44.

y(t)= t 2 y(t)= t 2 and x( t ) x( t ) is linear

45.

Write the parametric equations of a circle with center ( 0,0 ), ( 0,0 ), radius 5, and a counterclockwise orientation.

46.

Write the parametric equations of an ellipse with center ( 0,0 ), ( 0,0 ), major axis of length 10, minor axis of length 6, and a counterclockwise orientation.

For the following exercises, use a graphing utility to graph on the window [ 3,3 ] [ 3,3 ] by [ 3,3 ] [ 3,3 ] on the domain [0,2π) [0,2π) for the following values of a a and b b , and include the orientation.

{ x(t)=sin(at) y(t)=sin(bt) { x(t)=sin(at) y(t)=sin(bt)
47.

a=1,b=2 a=1,b=2

48.

a=2,b=1 a=2,b=1

49.

a=3,b=3 a=3,b=3

50.

a=5,b=5 a=5,b=5

51.

a=2,b=5 a=2,b=5

52.

a=5,b=2 a=5,b=2

Technology

For the following exercises, look at the graphs that were created by parametric equations of the form { x(t)=acos(bt) y(t)=csin(dt) . { x(t)=acos(bt) y(t)=csin(dt) . Use the parametric mode on the graphing calculator to find the values of a,b,c, a,b,c, and d d to achieve each graph.

53.
Graph of the given equations
54.
Graph of the given equations
55.
Graph of the given equations
56.
Graph of the given equations

For the following exercises, use a graphing utility to graph the given parametric equations.

  1. { x(t)=cost1 y(t)=sint+t { x(t)=cost1 y(t)=sint+t
  2. { x(t)=cost+t y(t)=sint1 { x(t)=cost+t y(t)=sint1
  3. { x( t )=tsint y( t )=cost1 { x( t )=tsint y( t )=cost1
57.

Graph all three sets of parametric equations on the domain [0,2π]. [0,2π].

58.

Graph all three sets of parametric equations on the domain [ 0,4π ]. [ 0,4π ].

59.

Graph all three sets of parametric equations on the domain [ 4π,6π ]. [ 4π,6π ].

60.

The graph of each set of parametric equations appears to “creep” along one of the axes. What controls which axis the graph creeps along?

61.

Explain the effect on the graph of the parametric equation when we switched sint sint and cost cost .

62.

Explain the effect on the graph of the parametric equation when we changed the domain.

Extensions

63.

An object is thrown in the air with vertical velocity of 20 ft/s and horizontal velocity of 15 ft/s. The object’s height can be described by the equation y( t )=16 t 2 +20t y( t )=16 t 2 +20t , while the object moves horizontally with constant velocity 15 ft/s. Write parametric equations for the object’s position, and then eliminate time to write height as a function of horizontal position.

64.

A skateboarder riding on a level surface at a constant speed of 9 ft/s throws a ball in the air, the height of which can be described by the equation y( t )=16 t 2 +10t+5. y( t )=16 t 2 +10t+5. Write parametric equations for the ball’s position, and then eliminate time to write height as a function of horizontal position.

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time t. t. Neglect air resistance.

65.

Find parametric equations that model the problem situation.

66.

Find all possible values of x x that represent the situation.

67.

When will the dart hit the ground?

68.

Find the maximum height of the dart.

69.

At what time will the dart reach maximum height?

For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.

70.

An epicycloid: { x(t)=14costcos(14t) y(t)=14sint+sin(14t) { x(t)=14costcos(14t) y(t)=14sint+sin(14t) on the domain [0,2π] [0,2π] .

71.

A hypocycloid: { x(t)=6sint+2sin(6t) y(t)=6cost2cos(6t) { x(t)=6sint+2sin(6t) y(t)=6cost2cos(6t) on the domain [0,2π] [0,2π] .

72.

A hypotrochoid: { x(t)=2sint+5cos(6t) y(t)=5cost2sin(6t) { x(t)=2sint+5cos(6t) y(t)=5cost2sin(6t) on the domain [0,2π] [0,2π] .

73.

A rose: { x(t)=5sin(2t)sint y(t)=5sin(2t)cost { x(t)=5sin(2t)sint y(t)=5sin(2t)cost on the domain [0,2π] [0,2π] .

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