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Variations on a Circle

Activity Description Activity Guide


Part 1: Drawing Circles in Three Coordinate Systems

(1) Write the Cartesian equation(s) for a circle centered at the origin with radius equal to 2. Use your graphing calculator to graph this circle. (Use the VARS key to help enter the equation for the negative branch of the graph). Comment on the relationship between the shape of this drawn circle and the proportionality of the window. If necessary, redraw your graph in a proportional window. Trace this circle and make note of the directionality of the trace cursor and the coordinates of key points on the graph.

(2) Write a set of parametric equations for a circle centered at the origin with radius equal to 2. Use your graphing calculator to graph this circle (on the same screen as the previous circle if your graphing calculator has this capability). Again, trace this circle, and make note of the directionality of the trace cursor and the coordinates of several key points on the graph.

(3) Write an equation for a circle centered at the origin with a radius of 2 using polar coordinates. Use your graphing calculator to graph this circle (on the same screen as the previous circle if your graphing calculator has such a capability). Again, trace this circle, and make note of the directionality of the trace cursor and the coordinates of several key points on the graph.

(4) Compare the starting point and direction of tracing for the 3 different coordinate systems, and compare the different coordinate representations of your key points. Discuss relationships among these coordinates.

(5) Compare and contrast various aspects and features of the 3 coordinate systems for the case of a circle centered at the origin.

Part 2: Translating Circles

(6) Write equations in each coordinate system for a circle centered at the point (3, -4) with radius equal to 5. Explain how you derived these equations.

Hint: In parametric coordinates, adjust the coordinates for x and y to translate the center. This yields x(t)=3+5cos(t) and y(t)=-4+5sin(t).

Hint: Recall that in polar coordinates, x = rcos() and y = rsin(), so that r2 = x2+y2. Substituting polar coordinate expressions for x-3 and y+4 yields r=6cos()-8sin().

(7) Use your graphing calculator to graph each of these circles. Revise your equations as necessary. Relate features of your equations to features of your graphs.

(8) Discuss advantages and disadvantages of each system for the case of a circle not centered at the origin..

(9) Use your graphing calculator to draw the following graphs (make use of all 3 coordinate systems to complete this task):

(10) Explain your approach to completing task (9).

Part 3: Drawing "Circles" of Few Points

(11) Again, draw a circle of radius 2 using parametric equations. Observe the view window. Comment on the step or pitch.

(12) Adjust the step or pitch of your window to draw the following polygons:

(13) Comment on how you went about generating these polygons, and on how the step size is related to the shape of each.

(14) Trace each of these graphs and observe the coordinates of the traced points. Relate these coordinates to your comments in task (13).

Part 4: Rotations and Other Transformations

(15) Try to draw the "upright" square in the first screenshot below without changing the equation used to generate the diamond above.

Hint: You can change the orientation by changing Tmin and Tmax in the window.

(16) Go back to the window settings you used to draw the diamond above. Examine your parametric equations for this figure. Now, adjust these equations to draw the "upright" square and the other quadrilaterals:

(17) Discuss your approach to completing task (15). Be sure to relate features of the graphs to those of the equations used to generate them.

(18) Now make adjustments to generate the following shapes:

(19) Explain how you went about deriving equations to generate the above shapes.



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Last modified on August 15, 2001.