Unit 4

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Unit 4: Polar Coordinates & Conic Sections

Lesson 1: Introduction to Polar Coordinates (极坐标)

Coordinate: how to find a point in a space

Rectangular coordinates (直角坐标) (x, y)

Polar coordinates (r, θ)

Rectangular to Polar:

Given y=f(x), find r=g(θ)

Substitute y=r sin⁡θ, x=r cos⁡θ

Given r=g(θ), find y=f(x)

Substitute r=√(x^2+y^2 ),tan⁡〖θ=y/x, sin θ=y/r,cos⁡〖θ=x/r, … 〗 〗

Lesson 2: Parabolas and Hyperbolas

Conics (圆锥曲线) general equation

Ax^2+Bxy+Cy^2+Dx+Ey+F=0

Parabolas: Either A=0, or C=0, not both

Regular and sideway, vertex (h, k)

y=a(x−h^2+k

x=a(y−k)^2+h

Rectangular Hyperbolas: AC=−1

Horizontal: (x−h^2/a^2 −(y−k)^2/b^2 =1

Vertical: (y−k)^2/a^2 −(x−h^2/b^2 =1

What is a and b on a graph?

Lesson 3: Circles and Ellipses

Circles: A=C

(x−h^2+(y−k)^2=r^2, or (x−h^2/r^2 +(y−k)^2/r^2 =1

Ellipses: AC>0, A≠C

Horizontal Ellipse a>b: (x−h^2/a^2 +(y−k)^2/b^2 =1

Vertical Ellipse a>b: (y−k)^2/a^2 +(x−h^2/b^2 =1

Major axis and Minor axis

Lesson 4: Polar Graphs and Rose Curves

Lesson 5: Polar graph and cardioids

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