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LG10 - Coggle Diagram
LG10
Conic sections
Equation of a circle: (x-h)^2+(y-k)^2=r^2, where (h,k) is the center and r is the radius. standard form
Standard form of a circle at (0, 0) and the radius is r: x^2+y^2=r^2.
For ellipses and hyperbolas, in standard form they are always equal to 1, and the length of the axes is all the way across (doubling a/b)
The parabola is one of a family of curve called conic sections, which are formed by the intersection of a double right cone and a plane. There are four types of conic sections: parabolas, circles, ellipses, and hyperbolas.
(x-h) --> h units right, and (x+h) --> h units left
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To tell the difference between a hyperbola and an ellipse: an ellipse uses addition while hyperbola equation uses subtraction
The major axis: The longer segment that the ellipse is symmetrical about. Like the diameter of the ellipse. The point where the major axis touches the ellipse are called vertices.
The minor axis: shorter segment of the ellipse that it is symmetrical about. The points where the minor axis is touching the ellipse are called co-vertices.
a: The length of the horizontal segment from the center of the shape
b: the length of the vertical segment from the center of the shape
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Systems of equations
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Solutions are (x,y) coordinates, could be 0, 1, or 2 solutions
Discriminant is negative (0 solutions), discriminant is positive (2 solutions), when the discriminant is 0 (1 solution)
There can be no solutions when something is never true. Ex: If something simplifies down to 4=6, then there's no solutions because 4 will never be equal to 6. This also means that the lines are parallel and will never meet or intersect, so there aren't any solutions.
There can also be infinitely many solutions if the lines are lying on top of each other or the equations comes down to something that is always true, like 0=0.
The solutions are where the lines intersect at a certain point, and can be solved for algebraically by substitution, elimination, setting it equal to each other, or graphing
Linear-quadratic systems: quad =- 0, you can use formula, factoring, square root method etc., and you can eliminate the y's by multiplying one equation by -1, then you get your solution(s) and plug in the x-intercepts to get y-values.
Quadratic-quadratic systems: can have 0, 1, or 2 solutions, 3 solutions is not an option unless a quadratic and a conic.
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