Graph each ellipse. ((x+3)^2)/(16)+((y+2)^2)/(4)=1 |...

Name: Problem 12, Chapter 13: Algebra A Combined Function
Uploaded: 2022-02-24T14:25:52-08:00
Duration: 1 min 18 s
Channel: Allison Knapp
Description: Graph each ellipse. ((x+3)^2)/(16)+((y+2)^2)/(4)=1

Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of all points for which the sum of the distances to two fixed points (foci) is constant. It is a closed, smooth curve that resembles an elongated circle. The properties of an ellipse, such as its orientation and dimensions, are governed by its equation, making it a fundamental object of study in analytic geometry.

Standard Form of Ellipse Equation

The standard form of an ellipse's equation is expressed as (x - h)²/a² + (y - k)²/b² = 1, where (h, k) represents the center of the ellipse, 'a' is the semi-major axis length, and 'b' is the semi-minor axis length. This form clearly indicates the translations from the origin and the scaling along the x and y axes, which are essential for graphing the ellipse accurately.

Center

The center of an ellipse is the point (h, k) in its standard equation and represents the midpoint of both the major and minor axes. This point serves as the reference for the ellipse’s symmetry and helps in determining how the graph is translated from the origin, which is crucial in plotting the ellipse correctly.

Semi-major and Semi-minor Axes

The values 'a' and 'b' in the standard form represent the lengths of the semi-major and semi-minor axes, respectively. The semi-major axis is the longer distance from the center to the ellipse along the direction of the ellipse's greatest width, while the semi-minor axis is the shorter distance perpendicular to the semi-major axis. These axes determine the shape and size of the ellipse.

Graphing Process

Graphing an ellipse involves identifying its center, determining the lengths of the semi-major and semi-minor axes, and plotting key points along these axes. Once these points are marked, the smooth, closed curve of the ellipse is drawn through them, maintaining the symmetry about the center. This analytical approach helps in creating an accurate visual representation of the ellipse.

Transcript

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