Solving Equations and Inequalities That Involve Conics.

You have a conic equation with exponents to the second power, and the equation of a line...How should you proceed to solve the system? What about if you have the equation of two conics? ...Should I graph them, or should I solve by substitution? ...Now, I I have a couple of inequalities with second degree exponents: How may I solve them? will I shade the interior of the graph or the exterior?

The promise you get with this lesson is that you will clarify these doubts. The lesson presents several examples that are solved graphically, and/or algebraically. You will find them very helpful to solve any future problem with similar characteristics. Go ahead, this may be the breakthrough you were looking for to be good at math! Try to highlight or write notes as you go through the lesson by using the MARKER TOOLS menu on the top of the screen. Some vocabulary terms are given at the bottom of the page and after the lesson. You may study them before going on the lesson for the first time.

Lesson's Content

Lesson In PDF Format (no animations)


Lesson's Glossary

Asymptote: If the graph of a function gets close to a line but never intersects with this then line is an asymptote.

Axis of Symmetry: A line on which a graph is reflected onto itself. 

Center of the ellipse: The point where the major and minor perpendicular axis intersect.

Center of the hyperbola: The point where the transversal and conjugate perpendicular axis intersect.

Completing the Square: Method that finds the constant term in an incomplete perfect square trinomial of a second degree equation to solve it.

Conjugate axis: The axis perpendicular to the transverse axis.

Conic section: A figure that is obtained slicing a double cone with a plane. (parabola, circle, hyperbola, and ellipse)

Ellipse All the points in a plane for which the distance to the foci (each focus) is constant.

Equation: A mathematical statement that has to expressions joined by the = sign. It has the right side of the equation (expression1) and the left side of the equation (expression 2)

Factoring: The process to brake a polynomial down into the product of several factors.

Factors: All whole numbers that are multiplied together to yield another number.

Factored Form: Any polynomial that is written as the product of polynomials of lower degree that may be obtained from the original polynomial.

Focus of an ellipse: Each one of the points in the major axis of the ellipse, from which sum of the distance from each to the set of points in the ellipse is constant.

Foci: Plural for focus.

Inequality: One expression that is different from another one.

Major axis: The major of the two perpendicular axis of symmetry in an ellipse, and in which are located the foci.

Minor axis: The smallest of the two perpendicular axis of symmetry in an ellipse.

Parabola: The u curved shape you get when graphing a quadratic equation.

Quadratic equation: An equation of the form

Quadratic formula: If ax2 + bx + c = and a is different from zero then the quadratic formula is given in terms of a, b, and c.

Quadratic function: Any function in the form of

f(x) = ax2 + bx + c where a is different from zero. The graph is a parabola and the largest exponent is 2.

Transverse axis: A segment in the hyperbola goes from vertex to vertex of the two branches, and it is contained in the line that goes through the foci.

Vertices of a hyperbola: Are identified as the endpoints in the line segment that is the transverse axis of the hyperbola.

Vertex: The lowest point for a parabola that opens up (minimum); the highest point for a parabola that opens down (maximum).

Vertex form of a quadratic function: The vertex form of a quadratic function is: f(x) = a(x-h)2 + k. The vertex coordinates are (h,k).


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