1 / 22

Plane Curves and Parametric Equations

Plane Curves and Parametric Equations. New Ways to Describe Curves (10.4). POD. Describe this curve. Is it a function?. POD. Describe this curve. A circle with center at (-2, 3), and radius 11. Is it a function? Nope. New vocabulary.

francesca-haney
Download Presentation

Plane Curves and Parametric Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Plane Curves and Parametric Equations New Ways to Describe Curves (10.4)

  2. POD Describe this curve. Is it a function?

  3. POD Describe this curve. A circle with center at (-2, 3), and radius 11. Is it a function? Nope.

  4. New vocabulary y = f(x) is a function, if f is a function. We know how this works. This is an example of a plane curve, but plane curves also include non-functions, like the conics we’ve been studying.

  5. New vocabulary Various examples P(a) and P(b) are the endpoints of curve C. If P(a) = P(b), the curve is a closed curve. If there is not overlap, it is a simple closed curve. Is there a way to describe these non-functions in terms of functions? (Of course there is, or we wouldn’t be doing this.) P(a) P(a) = P(b) P(b)

  6. New vocabulary Plane curve: a set C of ordered pairs (f(t),g(t)), where f and g are functions defined on interval I. In other words, a graph may not be a function, but we substitute the coordinates x and y with separate components that are functions of t– a mathematical sleight of hand. Then we specify an interval I to run t in.

  7. Parametric equations We use parametric equations to describe plane curves. The format is a bit different from the y = form we’re used to– we add a third variable and base the x and y functions on it. The curve C with parameter t: x = f(t) y = g(t) for t in interval I. The final result is a curve– which could be the same as curves we’ve seen– which runs in a particular direction (orientation).

  8. Parametric equations– use it The curve C with parameter t: x = f(t) y = g(t) for t in I. 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 a. On calculators, graph the curve, and determine its orientation. Change the T window and T step to see how the graph changes. b. Do this with a triple-column chart on the next slide. c. Combine them to find an equation in x and y (a more familiar form).

  9. Parametric equations– use it 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 a. Graph the curve, and show its orientation. We can do this easily on the graphing calculators. Change the MODE to “PARA” for parametric mode. The Y= window is different, and we include both functions.

  10. Parametric equations– use it 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 a. Plot the curve, and show its orientation. The window screen is also different– we include the final x and y dimensions, of course, but add the interval and increments for t.

  11. Parametric equations– use it 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 a. Plot the curve, and show its orientation. The final graph looks like something we’ve seen before. Why does it stop?

  12. Parametric equations– use it 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 b. Plot the curve, and show its orientation. t x y -1 -½ 0 ½ 1 3/2 2 It’s like there are two dependent variables (x and y) based on one independent variable (t). How does the curve “run”?

  13. Parametric equations– use it 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 c. Combine them to find an equation in x and y (a more familiar form). x = 2t t = x/2 y = t2 -1 y = (x/2)2 -1 y= ¼ x2 -1

  14. Parametric equations– use it 1. x = 2t y = t2 -1 -1 ≤ t ≤ 2 One curve can be expressed by an infinite number of parametric equations. x = t y = ¼ t2 – 1 -2 ≤ t ≤ 4 x = t3 y = (1/8) t6 – 1 -21/3 ≤ t ≤ 41/3 Try graphing them on calculators.

  15. Parametric equations– use it 2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P. Don’t graph, just think.

  16. Parametric equations– use it 2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P. Don’t graph, just think. Graph on calculators. Set a = 5.

  17. Parametric equations– use it 2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P.

  18. Parametric equations– use it 2. P(x, y) is x = a cos t and y = a sin t, for all real number values of t, and a >0. Describe the motion of P. Why look, a circle with radius a, centered on the origin. The curve follows a counter-clockwise rotation. Graph it on calculators to check. Radian or degree mode?

  19. Parametric equations– use it 3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation? x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2

  20. Parametric equations– use it 3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation? x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2 What does this curve look like?

  21. Parametric equations– use it 3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation? x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2 t x y -2 -1 0 1 2

  22. Parametric equations– use it 3. From p. 824, #4. Graph the parametric equation and give its orientation. What is the equation in x-y notation? x = t3 + 1 y = t3 – 1 -2 ≤ t ≤ 2 t3 = x – 1 y = (x – 1) – 1 y = x – 2

More Related