Vocabulary cardioid rose lemniscate spiral of Archimedes limacon ΄

Example 1 Graph Polar Equations by Plotting Points A. Graph r = 3 cos θ. Make a table of values to find the r-values corresponding to various values of  on the interval [0, 2π]. Round each r-value to the nearest tenth.

Example 1 Answer: Graph Polar Equations by Plotting Points Graph the ordered pairs (r,  ) and connect them with a smooth curve. It appears that the graph is a circle with center (1.5, 0) and radius 1.5 units.

Example 1 Graph Polar Equations by Plotting Points B. Graph r = 3 sin θ. Make a table of values to find the r-values corresponding to various values of  on the interval [0, 2π]. Round each r-value to the nearest tenth.

Example 1 Answer: Graph Polar Equations by Plotting Points Graph the ordered pairs and connect them with a smooth curve. It appears that the graph is a circle with center (0, 1.5) and radius 1.5 units.

Example 1 Graph r = –4 cos θ. A. B. C. D.

Key Concept 2

Example 2 Polar Axis Symmetry Use symmetry to graph r = 1 – 3cos θ. Replacing (r,  ) with (r, –  ) yields r = 1 – 3 cos(–  ). Because cosine is an even function, cos (–  ) = cos , so this equation simplifies to r = 1 – 3 cos . Because the replacement produced an equation equivalent to the original equation, the graph of this equation is symmetric with respect to the polar axis. Because of this symmetry, you need only make a table of values to find the r-values corresponding to  on the interval [0, π].

Example 2 Polar Axis Symmetry Plotting these points and using polar axis symmetry, you obtain the graph shown. Answer: This curve is called a limacon with an inner loop. ΄

Example 2 Use symmetry to graph r = cos . A. B. C. D.

Key Concept 3

Example 3 A. LIGHT TECHNOLOGY The area lit by two lights that shine down on a stage can be represented by the equation r = sin θ. Suppose the front of the stage faces due south. Graph the polar pattern of the two lights. Symmetry with Respect to the Line Because this polar equation is a function of the sine function, it is symmetric with respect to the line. Therefore, make a table and calculate the values of r on

Example 3 Symmetry with Respect to the Line Plotting these points and using symmetry with respect to the line, you obtain the graph shown. This curve is called a cardioid.

Example 3 Answer: Symmetry with Respect to the Line

Example 3 B. LIGHT TECHNOLOGY The area lit by two lights that shine down on a stage can be represented by the equation r = sin θ. Suppose the front of the stage faces due south. Describe what the polar pattern tells you about the two lights. Symmetry with Respect to the Line Answer: Sample answer: The polar pattern indicates that the lights will light up a large portion toward the back of the stage but will not light up very much past the edge of the stage into the audience.

Example 3 AUDIO TECHNOLOGY A microphone was placed at the front of a stage to capture the sound from the acts performing during the senior talent show. The front of the stage faces due south. The area of sound the microphone captures can be represented by r = sin . Describe what the polar pattern tells you about the microphone.

Example 3 A.The microphone will pick up a large portion of sound toward the back of the stage but not much from the front edge of the stage and audience. B.The microphone will pick up a large portion of sound toward the front of the stage and the audience but not much from the back of the stage. C.The microphone will pick up a large portion of sound on the right side of the stage and audience but not much from the left side. D.The microphone will pick up a large portion of sound on the left side of the stage and audience but not much from the right side.

Example 4 Symmetry, Zeros, and Maximum r-Values Use symmetry, zeros, and maximum r-values to graph r = 2 sin 2θ. Sketch the graph of the rectangular function y = 2 sin 2  on the interval This function is symmetric with respect to the polar axis and the line, so you can find points on the interval and then use symmetry to complete the graph.

Example 4 Symmetry, Zeros, and Maximum r-Values From the graph, you can see that |y| = 2 when and y = 0 when x = 0 and.

Example 4 Symmetry, Zeros, and Maximum r-Values Interpreting these results in terms of the polar equation r = 2sin 2 , we can say that |r| has a maximum value of 2 when  = and r = 0 when  = 0 and. Use these and a few additional points to sketch the graph of the function.

Example 4 Answer: Symmetry, Zeros, and Maximum r-Values Notice that polar axis symmetry can be used to complete the graph after plotting points on. This type of curve is called a rose.

Example 4 Determine the symmetry and maximum r-values of r = 5 sin 4  for 0 ≤ θ < π. A.symmetric to the line, |r| = 5 when B.symmetric to the polar axis, |r| = 5 when C.symmetric to the line, the polar axis, and the pole, | r | = 5 when D.symmetric to the line, the polar axis, and the pole, | r | = 5 when

Key Concept 5

Example 5 Identify and Graph Classic Curves A. Identify the type of curve given by r 2 = 8 sin 2θ. Then use symmetry, zeros, and maximum r-values to graph the function. The equation is of the form r 2 = a 2 sin 2 , so its graph is a lemniscate. Replacing (r,  ) with (–r,  ) yields (–r) 2 = 8 sin 2  or r 2 = 8 sin 2 . Therefore, the function has symmetry with respect to the pole.

Example 5 Identify and Graph Classic Curves The equation r 2 = 8 sin 2  is equivalent to r = which is undefined when 2 sin 2  < 0. Therefore, the domain of the function is restricted to the intervals Because you can use pole symmetry, you need only graph points in the interval. The function attains a maximum r-value of |a| or when and zero r-value when x = 0 and

Example 5 Identify and Graph Classic Curves Use these points and the indicated symmetry to sketch the graph of the function.

Example 5 Answer:lemniscates; Identify and Graph Classic Curves

Example 5 Identify and Graph Classic Curves B. Identify the type of curve given by r = 2θ, θ > 0. Then use symmetry, zeros, and maximum r-values to graph the function. The equation is of the form r = a  + b, so its graph is a spiral of Archimedes. Replacing (r,  ) with (–r, –  ) yields (–r) = 2(–  ) or r = 2 . Therefore, the function has symmetry with respect to the line However, since  > 0, this function will show no line symmetry.

Example 5 Identify and Graph Classic Curves Spirals are unbounded. Therefore, the function has no maximum r-values and only one zero when  = 0. Use points on the interval [0, 4π] to sketch the graph of the function.

Example 5 Answer:spiral of Archimedes; Identify and Graph Classic Curves

Example 5 Identify the type of curve given by r = 4 cos 6θ. A.cardioid B.lemniscate C.limacon D.rose ΄

End of the Lesson