Algebra 3

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Parabola

Description: the shape fits like a letter u and can face up, down, left or right

General formula:

Y=Ax2+bx=c

Examples:

3x2 + 2x -10 =0

-5y2+ 6y+7=0

HyperbolaDescription: The set of all points in a Cartesian plane such that the difference of their distances to two fixed points of the plane is equal to a positive constant.
Is an open curve, it has noend.

General Formula:

A2+By2+Cx+Dy+E=0

Examples:

-x2+y2+x+y+1

3x2-5y2+2x+y+1=0

Ellipse

Description: is a closed symmetric curve that results from cutting the surface of a coneby an oblique plane to the axis of symmetry

General Formula:

Horizontal Major Axis Vertical Major axis
x2a2 + y2b2= 1 x2b2+y2a2=1

Examples:(2,0) to (-2,0)

F'(-c, 0) y F(c, 0)

Straight Lines:

Description: The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, andis nothing else than the flow or run of the point which will leave from its imaginary moving some vestige in length, exempt of any width. The straight line is that which is equally extended betweenits points.

General Formula:
A straight line is defined by a linear equation whose general form is:
Ax + By + C = 0
Where A, B are not both 0. The coefficients A and B in the general equation arethe components of vector n = (A, B) normal to the line. The pair r = (x, y) can be looked at in two ways: as a point or as a radius-vector joining the origin to that point. The latter interpretationshows that a straight line is the locus of points r with the property
r·n = const.
That is a straight line is a locus of points whose radius-vector has a fixed scalar product with a given vector n,normal to the line. To see why the line is normal to n, take two distinct but otherwise arbitrary points r1 and r2 on the line, so that
r1·n = r2·n.
But then we conclude that:
(r1 - r2)·n = 0....
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