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Learn about different polar graphs, including limaçons (convex, dimpled, looped), lemniscates, rose curves, and cardioids. View compare and contrast between the 4 different types of polar graphs, and ...

Learn about different polar graphs, including limaçons (convex, dimpled, looped), lemniscates, rose curves, and cardioids. View compare and contrast between the 4 different types of polar graphs, and view my impressions on this final unit in pre-calculus honors.

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- 1. Nicholas Portugal Precalculus Honors Period A7 POLAR GRAPHS
- 2. During the last few weeks of pre-calculus, we learned a variety of different polar graphs through five packet activities using the TI-Nspire calculators. Polar graphs are essentially graphs on a circular coordinate plane compared to the conventional rectangular planes. Polar graphs can be represented using function graphs, which are comprised of sine waves that follow a distinct pattern to represent different components of the polar graph. While polar graphs can be represented using function graphs, as they both contain angular measurements, we also learned how to convert ‘polar coordinates’ (e.g. 5,90º) to ‘rectangular’ coordinates ([5 x cos 90],[5 x sin 90] 0,5) using respective sine and cosine formulas to differentiate between both ‘x’ and ‘y’ on the rectangular coordinate plane. OVERVIEW
- 3. Old French word for ‘snails.’ Bi-circular shape. Three types: Looped, dimpled, convex. r = a ± b (cos θ) or r = a ± b (sin θ) Looped: |a/b| < 1 Dimpled: 1 < |a/b| < 2 Convex: |a/b| ≥ 2 Circular: r = a (cos θ) or r = a (sin θ) Curves are formed as the circle rotates around another of equal radius. LIMAÇONS
- 4. Named for its flowery petals that extend from the origin. r = a [cos (nθ)] or r = a [sin (nθ)] Odd # of Petals: When n is odd (n). Curves formed as it increases from 0 to π. Even # of Petals: When n is even (2n). Curves formed as it increases from 0 to 2π. If n is even, the graph is symmetric about the x- axis, y-axis, and the origin. Depending on the n value, the graph will be shaped in a particular way. ROSES
- 5. Shaped like an infinity symbol or figure-eight. r2 = a2 [cos (2θ)] or r2 = a2 [sin (2θ)]. r = ±√a2 [cos (2θ)] or r = ±√a2 [sin (2θ)]. a ≠ 0 Graphs are generated as the angle increases gradually from 0 to 2π. Symmetrical across the x-axis, y-axis, and the origin. LEMNISCATES
- 6. A type of 1-cusped epicycloid limaçon that is created when a = b. r = a ± b (cos θ) or r = a ± b (sin θ). |a/b| = 1 Graphs generated as angle increases from 0 to 2π. Can be drawn by tracing the path of a point on a circle as the circle revolves around a fixed circle of equal radius. Tangents at the ends of any chord through the cusp point are at right angles and their length is 2a. CARDIOIDS
- 7. Limaçons • Bi-circular shape. • Three different types. Roses • Use the variable “n.” • Differs in the number of petals. Lemniscates • Shape never changes, only size. • Represents a figure-eight or infinity symbol graph. Cardioids • 1-cusped epicycloid limaçon. • Chord tangent lengths are perpendicular and are 2a in length. Similarities • Cardioids are a form of limaçons. • Some loops on inverted loop limaçons resemble petals from the rose curves. • Limaçon curves are formed by the circle rotating around another of equal radius, much like cardioids. • Lemniscates and roses are symmetrical by the x-axis, y-axis, and origin when the n-value is even for roses. • Limaçons and rose petals completely differ in shape depending on the equation. • All of these graphs are comprised of different curves that are represented accordingly on a function graph. COMPARE & CONTRAST
- 8. Overall, the polar unit was perhaps one of the most difficult units this entire year. I learned from the many mistakes I made along the way while completing the packets, and using the TI- Nspire calculators helped me to visualize how the graphs were drawn, and how they compared with function graphs. Using both limaçons and roses together was fascinating because they correlated so well with each other. I was able to learn about a new type of graph from this unit, as I previously only knew how to graph function graphs and rectangular graphs. After each packet I began to grasp polar graphs even better, though I did not particularly enjoy only learning through the lessons on the calculators. Finally, if there were one thing to change about this unit, I would have also provided supplemental lessons on the subject in addition to the packets to ensure greater understanding, especially for students taking calculus next year. SUMMARY

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