GRAPHING IN POLAR COORDINATES SYMMETRY

Transcription

1 GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis, x- axis, and origin. Do you recall how we could test the functions for symmetry? If not, here are the tests. 1. A graph has symmetry with respect to the y-axis if, whenever (x, y) is on the graph, so is the point (-x, y). 2. A graph has symmetry with respect to the origin if, whenever (x, y) is on the graph, so is the point (-x, -y). 3. A graph has symmetry with respect to the x-axis if, whenever (x, y) is on the graph, so is the point (x, -y). The big question is how do we test for symmetry of an equation in polar coordinates? Let us look at the following diagrams to determine the answer to this question. x-axis symmetry y-axis symmetry

2 symmetry about the origin So here are the symmetry tests for polar graphs. 1. Symmetry about the x-axis: If the point (r, ) lies on the graph, then the point (r, - ) or (-r, - ) also lies on the graph. 2. Symmetry about the y-axis: If the point (r, ) lies on the graph, then the point (r, - ) or (-r, - ) also lies on the graph. 3. Symmetry about the origin: If the point (r, ) lies on the graph, then the point (- r, ) or (r, + ) also lies on the graph. EXAMPLE 1: Identify the symmetries of the curve r = cos and then sketch the graph. (r, - ) r = cos (- ) r = cos (Remember that cosine is an even function.) x-axis symmetry: yes (-r, - ) -r = cos (- ) -r = cos r = -2-2 cos y-axis symmetry: no (-r, ) -r = cos r = -2-2 cos symmetry with respect to the origin: no Now, let us compare our findings with the graph of this function.

3 Notice that the graph's only symmetry is with respect to the x- axis, and this is what we determine with our testing. EXAMPLE 2: Identify the symmetries of the curve r = 2 + sin and then sketch the graph. (r, - ) r = 2 + sin (- ) r = 2 -sin (Remember that sine is an odd function.) x-axis symmetry: no (-r, - ) -r = 2 + sin (- ) -r = 2 - sin r = -2 + sin y-axis symmetry: no Is this a correct answer? No! Let us look at the graph of these two functions on the same coordinate axis.

4 r = 2 + sin is the purple graph r = sin is the teal graph We have the same graph, but they start in different places. Therefore, this function does have y-axis symmetry. Sometimes it is best to look at the graph of the polar function instead of trusting algebraic manipulation. EXAMPLE 3: Identify the symmetries of the curve r 2 = cos and then sketch the graph. (r, - ) r 2 = cos (- ) r 2 = cos (Remember that cosine is an even function.) x-axis symmetry: yes (-r, - ) (-r) 2 = cos (- ) r 2 = cos y-axis symmetry: yes (-r, ) (-r) 2 = cos r 2 = cos symmetry with respect to the origin: yes Now, let us compare our findings with the graph of this function.

5 Yes this graph does fit the results that we received from algebraic manipulation. SLOPES Now let us look at how to determine the slope of a polar curve r = f ( ). Remember that the slope of any curve is given by dy/ dx not dr/ d, so we will have to derive out the formula for dy/dx. Let x = r cos = f ( ) cos and y = r sin = f ( ) sin. If f is a differentiable function of, then so is x and y. When dx/ d 0, we can find dy/ dx from the parametric formula. EXAMPLE 4: Find the slope of the curve r = -1 + sin at. Now evaluate dy/dx at.

6 EXAMPLE 5: Find the slope of the curve r = cos 2 at / 2. Now evaluate dy/ dx at / 2. FINDING POINTS WHERE POLAR GRAPHS INTERSECT There are two types of intersection points. They are (1) simultaneous, and (2) non-simultaneous. Here is how your find both types of points. To find the simultaneous intersection points, set the two equations equal to each other and solve for. To find the non-simultaneous intersection points, graph both equations and determine where the graphs cross each other. EXAMPLE 6: Find the points of intersection (both types) of the pair of curves r = 1 + sin and r = 1 - sin. SIMULTANEOUS INTERSECTION POINTS 1 + sin = 1 - sin 2sin = 0 = 0 and = When = 0, then r = 1 + sin 0 = 1 (1, 0).

7 When =, then r = 1 + sin = 1 (1, ). NON-SIMULTANEOUS INTERSECTION POINTS Let us graph both equations on the same axis. r = 1 + sin is in purple r = 1 - sin is in teal Notice that the graphs cross each other at the point (0, 0), so this is the non-simultaneous intersection point. This is the only one. EXAMPLE 7: Find the points of intersection (both types) of the pair of curves r = cos and r = 1 - cos. SIMULTANEOUS INTERSECTION POINTS

8 NON-SIMULTANEOUS INTERSECTION POINTS r = cos is in purple r = 1 - cos is in teal The graphs cross each other at the origin, so the only nonsimultaneous intersection point is (0, 0). EXAMPLE 8: Find the points of intersection (both types) of the pair of curves r 2 = cos 2 and r 2 = sin 2. SIMULTANEOUS INTERSECTION POINTS

9 NON-SIMULTANEOUS INTERSECTION POINTS r 2 = sin 2 is in purple r 2 = cos 2 is in teal The only non-simultaneous intersection point for these two graphs is the origin, (0, 0). I have discussed three major topics in this set of supplemental notes. The first was how to determine the symmetry of a polar graph. When looking at some examples, we concluded that we would sometimes have to look at the graph of the equation. The use of symmetry will be important when we start to determine the area inside the curve. The second topic that I discussed is the slope of a polar curve. This is an application of the derivative of a parametric curve. Finally, I talked about how to find the two types of intersection points. This will be useful when we start to determine the area between two curves