Warm Up—begin after the Quiz Find the absolute max & min value in and where they occur: 2. Determine if the graph of the function is symmetric about x, y or the origin: 3. Use trig ID’s to simplify: 4. 5. Hint: Use graph and table, see if you can answer in Radians Hint: symmetry about… x- axis (x,-y) when (x,y) y-axis f(-x) = f(x) Origin f(-x) = -f(x) Hint: use the sum and difference formulas for sine and cosine…remember those?

Graphs of Polar Equations Section 6.5 Graphs of Polar Equations

Graphs of Polar Equations Polar curves are actually just special cases of parametric curves. Keep in mind that polar curves are graphed in the (x, y) plane, despite the fact that they are given in terms of r and θ. That is why the polar graph of r = 4 cos θ is a circle rather than a cosine curve. Function Mode Polar Mode In function mode, points are determined by a vertical coordinate that changes as the horizontal coordinate moves left to right. In polar mode, points are determined by a directed distance from the pole that changes as the angle sweeps around the pole.

Checking for Symmetry Since modern graphing calculators produce these graphs so easily in polar mode, we are frankly going to assume that you do not have to sketch them by hand. Instead we will concentrate on analyzing the properties of the curves. Recall: we learned algebraic tests for rectangular equations in section 1.2 to check for symmetry: x- axis show (x,-y) when (x,y) y-axis show f(-x) = f(x) Origin show f(-x) = -f(x) OR Even show f(-x) = f(x) Odd show f(-x) = -f(x)

Symmetry Three types of symmetry figures to be considered will have are: The x-axis (polar axis) as a line of symmetry. The y-axis (the line θ = π/2) as a line of symmetry, (90°). The origin (the pole) as a point of symmetry, (looks the same upside-down and right-side up). All three algebraic tests for symmetry in polar forms require replacing the pair (r, θ), which satisfies the polar equation, with another coordinate pair and determining whether it also satisfies the polar equation.

Symmetry Test for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation. To Test for Symmetry Replace With About the x-axis, (r, θ) (r, -θ) or (-r, π – θ) About the y-axis, (r, θ) (-r, -θ) or (r, π – θ) About the pole, (origin) (r, θ) (-r, θ) or (r, θ + π)

Symmetry Test for Polar Graphs EXAMPLE 1 Testing for Symmetry Use the symmetry tests to prove that the graph of r = 4 sin 3θ is symmetric about the y-axis. First look at the graph to verify what you are trying to prove: If your graph doesn’t look like this, try resetting your calculator. As you can see the graph looks to be symmetric about y.

Analyzing Polar Graphs We analyze graphs of polar equations in much the same way that we analyze the graphs of rectangular equations. Looking at example 1, r = 4 sin 3θ, Trace can be used to help determine the range (max/min y-value) of this polar function. It can be shown that -4 ≤ r ≤ 4. Usually, we are more interested in the maximum value of |r| rather than the range of r in polar equations. In this case |r| ≤ 4 so we can conclude that the graph is bounded.

Maximum r-value Alright, so who wondered where π/6 and π/2 come from?? A maximum value for |r| is a maximum r-value for a polar equation. A maximum r-value occurs at a point on the curve that is the maximum distance from the pole. In r = 4 sin 3θ, a maximum r-value occurs at (4, π/6) and (-4, π/2). In fact, we get a maximum r-value at every (r, θ) which represents the tip of one of the three petals. Alright, so who wondered where π/6 and π/2 come from?? Using the FUNC graph, you can see that y=4, when x=.532 (aprox.)  You can use: CALC MAX/MIN

Maximum r-value To find the maximum r-values we must find maximum values |r| as opposed to the directed distance r. Example 2 shows one way to find maximum r-values graphically. EXAMPLE 2 Finding Maximum r-Values Find the maximum r-value of r = 2 +2 cos θ. Because we are interested in values of r, use the graph of the rec. eqn in function mode. You will see that the max value of r or y, is 4, which occurs when x or π, is a multiple of 2 π

Maximum r-value EXAMPLE 3 Finding Maximum r-Values Identify the points on the graph of r = 3 cos 2θ for 0 ≤ θ ≤ 2π that give maximum r-values. Polar Function Abs val. Function Using trace we can see that there are 4 points that have a max distance of 3 from the pole

Rose Curves The curve in Example 1 is a 3-petal rose curve and the curve in Example 3 is a 4-petal rose curve. The graphs of the polar equations r = a cos nθ and r = a sin nθ, where n is an integer greater than 1, are rose curves. If n is odd, then there are n petals, and if n is even there are 2n petals. r = 4 sin 3θ r = 3 cos 2θ

Rose Curves Graphs of Rose Curves The graphs of r = a cos nθ and r = a sin nθ, where n > 1 is an integer, have the following characteristics: Domain: All reals Range: [-|a|, |a|} Continuous Symmetry: n even, symmetric about x-, y-axis, origin n odd, r = a cos nθ symmetric about x-axis n odd, r = a sin nθ symmetric about y-axis Bounded Maximum r-value: |a| No asymptotes Number of petals: n, if n is odd 2n, if n is even

Rose Curves EXAMPLE 4 Analyzing a Rose Curve Analyze the graph of the rose curve r = 3 sin 4θ. Domain: Range: Continuous: Symmetry: Bounded: Max r-value: Asymptotes: Number of Petals:

Limaçon Curves (“LEE-ma-sohn”) The limaçon curves are graphs of polar equation of the form r = a ± b sin θ and r = a ± b cos θ where a > 0 and b > 0. Limaçon is Old French for “snail”. There are four different shapes of limaçons. Inner Loop: When a/b < 1 as in r = 1 + 2 sin θ Cardioid: When a/b = 1 as in r = 2 + 2 sin θ Dimpled: When 1 < a/b < 2 as in r = 3 + 2 cos θ Convex: When a/b ≥ 2 as in r = 2 + cos θ

Limaçon Curves Graphs of Limaçon Curves The graphs r = a ± b sin θ and r = a ± b cos θ , where a > 0 and b > 0, have the following characteristics: Domain: All reals Range: [a – b , a + b] Continuous Symmetry: r = a ± b sin θ, symmetric about the y-axis r = a ± b cos θ, symmetric about the x-axis Bounded Maximum r-value: a + b No asymptotes

Limaçon Curves EXAMPLE 5 Analyzing a Limaçon Curve Analyze the graph of r = 3 – 3 sin θ cardiod Domain: Range: Continuous: Symmetry: Bounded: Max r-value: Asymptotes:

Other Polar Curves-Spirals All the polar curves we have graphed so far have been bounded. The spiral in Example 6 is unbounded. EXAMPLE 6 Analyzing the Spiral Archimedes Analyze the graph of r = θ for [0, 10π] Note: these are not bounded We choose 10π, so that we can see more spirals, Notice that the spiral crosses the axis 10 times! Domain: Range: Continuous: Symmetry: Bounded: Max r-value: Asymptotes:

Other Polar Curves-Lemniscate From Latin meaning, “decorated with ribbons” The lemniscate curves are graphs of polar equation of the form r2 = a2 sin 2θ for [0, 2π] Example Graph: or

Other Polar Curves-Lemniscate Example 7 Analyzing a Lemniscante Curve Analyze the graph of r2 = 4 cos 2θ for [0, 2π] Domain: Range: Continuous: Symmetry: Bounded: Max r-value: Asymptotes:

Project Due 3/30 or 4/2 (Friday/Monday) Homework: Pg 540: 3-43 (o), 49, 51 Project Due 3/30 or 4/2 (Friday/Monday) It is still due, even if you are out for a school related absence!! Test 4/2 or 4/3 (Monday/Tuesday) Review due test day, go to Help on-line for selected solutions ReQuiz next Tuesday or Thursday Morning!!