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Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
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Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids

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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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