Parametric Equations of Line Segments: what is the slope? what is the y-intercept? how do we find the parametric eqtn of a given line segment?

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1 Shears Math 122/126 Parametric Equations Lecture Notes Use David Little's program for the following: Parametric Equations in General: look at default in this program, also spiro graph Parametric Equations of Line Segments: what is the slope? what is the y-intercept? how do we find the parametric eqtn of a given line segment? Parametric Equations of Circles: You already know one since we defined the trig functions on a unit circle.

2 Parametric Equations of Ellipses: Let's elongate the circle. Let's prove we got an ellipse.

3 Parametric Equations of Hyperbolas: Similar to ellipse except... Where are the formulas in your calculator? Optional: Parametric Equations of Parabolas

4 Shears Math 122/126 Unit 8 Act 4 Parametric Equations Activity Your job is to come up with the parametric equations for your assigned parts of this face. Let x and y be functions of t, where t starts at 0 and ends at 2π. For all parts of the graph right and left are with respect to the face (opposite of your right and left as you lookat the face). The outline is a circle. The right eye is an ellipse. The left eye is part of a hyperbola. (See more information on the left eye below.) The mouth is part of an ellipse. The right iris is a circle. The top of the nose, bottom of the nose, and both eye brows are straight line segments. Check your answers in your graphing calculator. For best results use tmin=0, tmax=2π, tstep=0.05, xmin = -6, xmax = 6, xscl = 1, ymin = -4, ymax=4, and yscl=1 in your TI calculator. If you are using a different graphing devise, experiment. The left eye is part of the top part of a hyperbola with center at (1.5,0.75) and vertex at (1.5,1.25). The x-coordinate goes from 0.5 to 2.5 and the y-coordinates range from 1.25 to about The guide box that goes with the left eye has a width of 2 units. You may be asking, "What do you mean by the guide box?" I am talking about that little box that you draw when you are graphing a hyperbola to help you draw a nicer graph. The vertices of the graph touch the box and the asymptotes of the graph go through the corners of the box. You may find it helpful to play with the Investigating Parametric Curves applet by David Little.

5 Shears Math 122 / 126 Unit 8 Skills 4 Parametric Equations Attach paper as needed. 1.) Find a rectangular equation equivalent to the following set of parametric equations: x = sec t, y = tan t, 0< t < 2.π 2.) Find a set of parametric equations equivalent to the rectangular equation: x 2 + y 2 = 4. Do not use x(t) = t. Do not use y(t) = t. Make sure you include restrictions on t. 3.) Graph the plane curve given by the parametric equations: Mark your scales. x(t) = 2 -t, y(t) = -2 t ; -3 t 3 4.) Come up with a set of parametric equations that describes the line segment that starts at (-2, 5) when t = 0 and ends at (2, 3) when t = 4. 5.) Sketch the graph of x(t) = 6tan 2 (t) + 4, y(t) = 2 tan(t) + 2 where t is in [-Mark your scales.

6 6.) Match the rectangular equations to their equivalent set of parametric equations. Assume t for equations involving sine and cosine and t for equations involving secant and tangent. a) x(t) = 2cos t 4, y(t) = 4sin t + 1 b) x(t) = 2cos t + 4, y(t) = 4sin t 1 x 4 y c) x(t) = 2sec t + 4, y(t) = 4tan t 1 d) x(t) = 2sec t 4, y(t) = 4tan t + 1 e) x(t) = 2tan t + 4, y(t) = 4tan t 1 f) x(t) = 2tan t 4, y(t) = 4tan t ) Fill in the chart and graph the plane curve given by the parametric equations and give the equivalent rectangular equation. x = sec 2 t 4, y = tan t + 1; 0 t t x y 0 rectangular equation: