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7.5 powerpoint presentation-original

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  1. 10.6 Parametric Equations

    Slide 1 - 10.6 Parametric Equations

  2. The equation of a graph in two dimensions is said to be in parametric form if each coordinate of a general point P(x, y) is expressed in terms of a third variable.  The third variable, usually denoted by a letter, is called a parameter.

    Slide 2 - The equation of a graph in two dimensions is said to be in parametric form if each coordinate of a general point P(x, y) is expressed in terms of a third variable. The third variable, usually denoted by a letter, is called a parameter.

    • Definition of a Plane Curve
    • If f and g are continuous functions of t, the set of ordered pairs (f(t),g(t)) is a plane curve C. The equations x=f(t) and y=g(t) are parametric equations for C, and t is the parameter.
  3. We can program a robot’s path by using a parameter.  To do this, we need two equations: one for the robot’s

    Slide 3 - We can program a robot’s path by using a parameter. To do this, we need two equations: one for the robot’s

    • x-coordinate, given by x=f(t)
    • and one for its y-coordinate, given by y=g(t).
    • The equation for x describes the robot’s east-west motion, and the equation for y describes its north-south motion. The two equations for x and y are called parametric equations with parameter t.
  4. Example 1

    Slide 4 - Example 1

    • Suppose we program the robot to follow a path defined by
    • for
    • where t is the time in minutes
    • a. Create a table of x and y values
    • t
    • x
    • y
    • b. Plot the points generated in part (a), and sketch the path followed by the robot (otherwise known as the “plane curve”). Indicate the orientation of the curve with arrows.
  5. c.   We can use substitution to rewrite our formula for x in terms of y.  The path followed by the robot can be written:

    Slide 5 - c. We can use substitution to rewrite our formula for x in terms of y. The path followed by the robot can be written:

    • Since the parameter t can be so easily eliminated from our equations, you may wonder why we should bother using it in the first place. One reason is that it can be useful to know not only where the robot is but also when it gets there. Taken together, the values of x and y tell us where the robot is while the parameter t tells us when it gets there. Rewriting the equation using only x and y is called “eliminating the parameter.”
  6. Example 2

    Slide 6 - Example 2

    • Suppose we program the robot to follow a path defined by
    • where t is the time in minutes
    • a. Create a table of x and y values.
    • t
    • x
    • y
    • b. Sketch the curve represented by the parametric equations
    • (indicate the orientation of the curve)
    • c. Eliminate the parameter and write the corresponding rectangular equation
  7. Example 3

    Slide 7 - Example 3

    • Suppose we program the robot to follow a path defined by
    • where t is the time in minutes
    • a. Create a table of x and y values.
    • t
    • x
    • y
    • b. Sketch the curve represented by the parametric equations
    • (indicate the orientation of the curve)
    • c. Eliminate the parameter and write the corresponding rectangular equation
  8. Example 4

    Slide 8 - Example 4

    • Suppose we program the robot to follow a path defined by
    • where t is the time in minutes
    • a. Create a table of x and y values.
    • t
    • x
    • y
    • b. Sketch the curve represented by the parametric equations
    • (indicate the orientation of the curve)
    • c. Eliminate the parameter and write the corresponding rectangular equation
  9. Example 5

    Slide 9 - Example 5

    • Suppose we program the robot to follow a path defined by
    • where t is the time in minutes
    • a. Create a table of x and y values.
    • t
    • x
    • y
    • b. Sketch the curve represented by the parametric equations
    • (indicate the orientation of the curve)
    • c. Eliminate the parameter and write the corresponding rectangular equation *Remember that
  10. Example 6

    Slide 10 - Example 6

    • Eliminate the parameter and obtain the standard form of the rectangular equation.
    • Circle:
  11. Example 7

    Slide 11 - Example 7

    • Use the result of Example 6 to find a set of parametric equations for a circle with center (-3, 2) and radius 5
  12. Example 8

    Slide 12 - Example 8

    • Find a set of parametric equations for the rectangular equation for using
  13. Example 9

    Slide 13 - Example 9

    • Use a graphing utility to sketch a graph of the curve represented by the parametric equations.
    • (*NOTE: You will have to adjust your window to get a good picture! Record the window you select.)
    • Prolate Cycloid
    • WINDOW
    • Tmin
    • Tmax
    • Tstep
    • Xmin
    • Xmax
    • Xscl
    • Ymin
    • Ymax
    • Yscl