Graphing Parametric Equations:

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Presentation transcript:

Graphing Parametric Equations:

Graphing Parametric Equations: Suppose that the system of parametric equations given below represents the path that a figure-skater carves on the ice:

Graphing Parametric Equations: Suppose that the system of parametric equations given below represents the path that a figure-skater carves on the ice: Let’s sketch a graph of the figure-skater’s movement. We can start by constructing a table of values so that we can find ordered pairs on the graph of the system of parametric equations.

Graphing Parametric Equations: To construct a table of values for the system

Graphing Parametric Equations: To construct a table of values for the system let’s choose values for t

Graphing Parametric Equations: To construct a table of values for the system let’s choose values for t

Graphing Parametric Equations: To construct a table of values for the system let’s choose values for t, find corresponding values of x(t)

Graphing Parametric Equations: x(t) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t)

Graphing Parametric Equations: x(t) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t) and y(t),

Graphing Parametric Equations: x(t) y(t) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t) and y(t),

Graphing Parametric Equations: x(t) y(t) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t))

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t))

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t)) on a coordinate plane.

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) To construct a table of values for the system let’s choose values for t, find corresponding values of x(t) and y(t), and then plot ordered pairs (x(t), y(t)) on a coordinate plane.

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) )

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) Let’s start with t = 0.

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) Let’s start with t = 0.

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) Let’s start with t = 0. Then

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) Let’s start with t = 0. Then

Graphing Parametric Equations: x(t) y(t) (x(t), y(t) ) Let’s start with t = 0. Then and

Graphing Parametric Equations: x(t) y(t) 8 (x(t), y(t) ) Let’s start with t = 0. Then and

Graphing Parametric Equations: x(t) y(t) 8 (x(t), y(t) ) Let’s start with t = 0. Then and So (0, 8) is on the graph of this system.

Graphing Parametric Equations: x(t) y(t) 8 (x(t), y(t) ) (0, 8) Let’s start with t = 0. Then and So (0, 8) is on the graph of this system.

Graphing Parametric Equations: x(t) y(t) 8 (x(t), y(t) ) (0, 8) Let’s start with t = 0. Then and So (0, 8) is on the graph of this system. Let’s plot this point on our coordinate plane.

Graphing Parametric Equations: x(t) y(t) 8 (x(t), y(t) ) (0, 8)

Graphing Parametric Equations: x(t) y(t) 8 (x(t), y(t) ) (0, 8) Now let’s take t = 1.

Graphing Parametric Equations: 1 x(t) y(t) 8 (x(t), y(t) ) (0, 8) Now let’s take t = 1.

Graphing Parametric Equations: 1 x(t) y(t) 8 (x(t), y(t) ) (0, 8) Now let’s take t = 1. Then

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 (x(t), y(t) ) (0, 8) Now let’s take t = 1. Then

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 (x(t), y(t) ) (0, 8) Now let’s take t = 1. Then and

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) Now let’s take t = 1. Then and

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) Now let’s take t = 1. Then and So is on the graph of this system. (4, 5.7)

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 1. Then and So is on the graph of this system. (4, 5.7)

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 1. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane (4, 5.7)

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 1. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (4, 5.7)

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7)

Graphing Parametric Equations: 1 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2.

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2.

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2. Then

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2. Then

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2. Then and

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2. Then and

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (4, 5.7) Now let’s take t = 2. Then and So is on the graph of this system. (0, 0)

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 2. Then and So is on the graph of this system. (0, 0)

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 2. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane (0, 0)

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 2. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (0, 0)

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7)

Graphing Parametric Equations: 1 2 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3.

Graphing Parametric Equations: 1 2 3 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3.

Graphing Parametric Equations: 1 2 3 x(t) 4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3. Then

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3. Then

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3. Then and

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3. Then and

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) Now let’s take t = 3. Then and So is on the graph of this system. (-4, -5.7)

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 3. Then and So is on the graph of this system. (-4, -5.7)

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 3. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane (-4, -5.7)

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 3. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (-4, -5.7)

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7)

Graphing Parametric Equations: 1 2 3 x(t) 4 -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 4.

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 4.

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then and

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then and

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then and So is on the graph of this system. (0, -8)

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then and So is on the graph of this system. (0, -8)

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane (0, -8)

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 4. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (0, -8)

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7)

Graphing Parametric Equations: 1 2 3 4 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5.

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5.

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5. Then

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5. Then

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5. Then and

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5. Then and

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) Now let’s take t = 5. Then and So is on the graph of this system. (4, -5.7)

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 5. Then and So is on the graph of this system. (4, -5.7)

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 5. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane (4, -5.7)

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 5. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (4, -5.7)

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7)

Graphing Parametric Equations: 1 2 3 4 5 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6.

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6.

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then and

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then and

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then and So is on the graph of this system. (0, 0)

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then and So is on the graph of this system. (0, 0)

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 6. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (0, 0)

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7)

Graphing Parametric Equations: 1 2 3 4 5 6 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7.

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7.

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7. Then

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7. Then

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7. Then and

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7. Then and

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) Now let’s take t = 7. Then and So is on the graph of this system. (-4, 5.7)

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) (-4, 5.7) Now let’s take t = 7. Then and So is on the graph of this system. (-4, 5.7)

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) (-4, 5.7) Now let’s take t = 7. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane (-4, 5.7)

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) (-4, 5.7) Now let’s take t = 7. Then and So is on the graph of this system. Let’s plot this point on our coordinate plane and connect our points. (-4, 5.7)

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) (-4, 5.7)

Graphing Parametric Equations: 1 2 3 4 5 6 7 x(t) -4 y(t) 8 5.7 -5.7 (x(t), y(t) ) (0, 8) (0, 0) (0, -8) -8 (4, 5.7) (-4, -5.7) (4, -5.7) (-4, 5.7) Finally, we can finish our graph. We can now see that the figure skater carved a figure-eight on the ice.