Conic Sections and Parabolas

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Presentation transcript:

Conic Sections and Parabolas 10.1 Conic Sections and Parabolas

Quick Review

Quick Review Solutions

What you’ll learn about Conic Sections Geometry of a Parabola Translations of Parabolas Reflective Property of a Parabola … and why Conic sections are the paths of nature: Any free-moving object in a gravitational field follows the path of a conic section.

Parabola A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.

A Right Circular Cone (of two nappes)

Conic Sections and Degenerate Conic Sections

Conic Sections and Degenerate Conic Sections (cont’d)

Second-Degree (Quadratic) Equations in Two Variables

Structure of a Parabola

Graphs of x2=4py

Parabolas with Vertex (0,0) Standard equation x2 = 4py y2 = 4px Opens Upward or To the Right Downword or to the left Focus (0,p) (p,0) Directrix y = -p x = -p Axis y-axis x-axis Focal length p p Focal width |4p| |4p|

Graphs of y2 = 4px

Example Finding an Equation of a Parabola

Example Finding an Equation of a Parabola

Parabolas with Vertex (h,k) Standard equation (x-h)2 = 4p(y-k) (y-k)2 = 4p(x-h) Opens Upward or downward To the right or to the left Focus (h,k+p) (h+p,k) Directrix y = k-p x = h-p Axis x = h y = k Focal length p p Focal width |4p| |4p|

Example Finding an Equation of a Parabola

Example Finding an Equation of a Parabola