Polar Curve Activity. Directions

Transcription

1 Polar Curve Activity Name: Activity Objectives Students will be able to use the graphing calculator to graph curves in polar form. Students will recognize the graph that is produced from an equation in polar form. Students will be able to analyze the characteristics of a graph given its equation in polar form. Directions In this activity, you will use your graphing calculator to graph equations given in polar form. Remember the following when using the calculator. 1. Change the mode to Pol. 2. Most of the equations contain a trigonometric function. They are periodic functions. Radian mode will allow you to use smaller values of!. 3. Graph using the bubble to draw you curves. This will help with the analysis of the curves. 4. Set your window to the same dimensions as the grid given for each set of graphs. While graphing the curves in this activity, in addition to answering the questions in each section, keep in mind the following characteristics. - domain and range - symmetry - how long it takes to graph a complete curve Two of the basic forms of a polar curve are given by r(") = a + bcos(n") and r(") = a + bsin(n"). By changing the values of a, b, and n, different polar curves can be generated.

2 Circles r 1 = 2 cos(! ), r 2 = 3 cos(! ), r 3 = " 4 cos(! ), r 4 = " 5 cos(! ) 2. In the equation r(") = a + bcos(n"), what is the value of a for each of the equations in #1? What is the value of n? a = n = 3. What effect does b have on the graph of the circle? 4. Replace cosine with sine in each of the equations in #1 and graph them on the calculator. - Page 2 of 7 -

3 Rose Curves r 1 = 4cos(! ), r 2 = 4cos(2! ), r 3 = 4cos(3! ), r 4 = 4cos(4! ) 2. In the equation r(") = a + bcos(n"), what is the value of a for each of the equations in #1? What is the value of b? a = b = 3. How does the value of n determine the number of leaves? 4. Change the value of b in each of the equations in #1 and graph them on the calculator. What effect does the value of b have on the leaves of the rose? 5. Graph r = 5 cos(3! ) and r = "5 cos(3! ) on the calculator. What effect does the sign of b have on the curve? 6. Replace cosine with sine in each of the equations in #1 and graph them on the calculator. - Page 3 of 7 -

4 Limaçon Curves r 1 = 1+ 2cos(! ), r 1 = 2 + 4cos(! ), r 3 = 1" 3cos(! ), r 4 = 2 " 5cos(! ) 2. In the equation r(") = a + bcos(n"), what is the value of n for each of the equations in #1? 3. How does the absolute value of a compare to the absolute value of b? 4. How do the absolute values of a and b effect the graph? 5. What effect does the sign of b have on the graph? 6. Replace cosine with sine in each of the equations in #1 and graph them on the calculator. - Page 4 of 7 -

5 Cardioid Curves r 1 = 1+ cos(! ), r 1 = 2 + 2cos(! ), r 3 = 3 + 3cos(! ), r 4 = 4 + 4cos(! ) 2. In the equation r(") = a + bcos(n"), what is the value of n for each of the equations in #1? 3. How does the absolute value of a compare to the absolute value of b? 4. How do the absolute values of a and b effect the graph? 5. Graph r = 3 + 3cos(! ) and r = 3 " 3cos(! ) on the calculator. What effect does the sign of b have on the curve? 6. Replace cosine with sine in each of the equations in #1 and graph them on the calculator. - Page 5 of 7 -

6 Lemniscate Curves You will have to graph each curve in two pieces. Use a small! -step to see the entire graph. 2 2 r = 9cos(2! ), r = 16cos(2! ) 2. Lemniscate curves are of the form r 2 = a 2 cos(2"). How does the value of a effect the curve? 3. Replace cosine with sine in each of the equations in #1 and graph them on the calculator. - Page 6 of 7 -

7 Spiral of Archimedes 1. Graph r =! for "! 0 and "! 0. Sketch them separately in the grids below. - Page 7 of 7 -