How Do You Find The Path• Plot points for various values of t, being careful to notice what range of values t should assume• Eliminate the parameter and find one equation relating x and y• Use the TI82/83 in parametric mode
Plotting Points• Note the direction the path takes• Use calculus to find – maximum points – minimum points – points where the path changes direction• Example: Consider the curve given by x = t + 1, y = 2t , − 5 ≤ t ≤ 5 2
Consider x = t 2 + 1, y = 2t , − 5 ≤ t ≤ 5• The parameter t ranges from -5 to 5 so the first point on the path is (26, -10) and the last point on the path is (26, 10)• x decreases on the t interval (-5,0) and increases on the t interval (0,5). (How can we tell that?)• y is increasing on the entire t interval (-5,5). (How can we tell that?)
Note Further x = t + 1, y = 2t , − 5 ≤ t ≤ 5 2• x has a minimum when t=0 so the point farthest to the left on the path is (1,0).• x is maximal at the endpoints of the interval [-5,5], so the points on the path farthest to the right are the starting and ending points, (26, -10) and (26,10).• The lowest point on the path is (26,-10) and the highest point is (26,10).
Eliminate the ParameterStill use x = t + 1, y = 2t , − 5 ≤ t ≤ 5 2Solve one of the equations for t Here we get t=y/2Substitute into the other equation Here we get x = ( y / 2) + 1 or x = ( y / 4) + 1 2 2
Summary• Use parametric equations for a curve not given by a function.• Use parametric equations to describe paths.• Each coordinate requires one function.• The parameter may be time, angle, or something else altogether...
A particular slide catching your eye?
Clipping is a handy way to collect important slides you want to go back to later.