# The parabola

Parabolas In this section we want to look at the graph of a quadratic function. This reflective property is the basis of many practical uses of parabolas. Again, be careful to get the signs correct here! The same effects occur with sound and other forms of energy. The " latus rectum " is the chord of the parabola which is parallel to the directrix and passes through the focus.

Now, the left part of the graph will be a mirror image of the right part of the graph.

Conversely, light that originates from a point source at the focus is reflected into a parallel " collimated " beam, leaving the parabola parallel to the axis of symmetry.

So what is this going to be equal to? Here are the vertex evaluations. Finding intercepts is a fairly simple process. Here are some examples of parabolas. Now, there are two forms of the parabola that we will be looking at. It was just included here since we were discussing it earlier. In this orientation, it extends infinitely to the left, right, and upward.

Be very careful with signs when getting the vertex here. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles.

They are frequently used in physicsengineeringand many other areas. So, when we are lucky enough to have this form of the parabola we are given the vertex for free. Example 2 Sketch the graph of each of the following parabolas. Here is a sketch of the graph.

Here are the evaluations for the vertex. So, we need to take a look at how to graph a parabola that is in the general form. The point on the parabola that intersects the axis of symmetry is called the " vertex ", and is the point where the parabola is most sharply curved.

Unlike the previous form we will not get the vertex for free this time. Parabolas have the property that, if they are made of material that reflects lightthen light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs.

The focus does not lie on the directrix. Now, the vertex is probably the point where most students run into trouble here. If we are correct we should get a value of Now we have to be careful. Here is the vertex for a parabola in the general form. This is nothing more than a quick function evaluation.

So, to get that we will first factor the coefficient of the x2 term out of the whole right side as follows. For other uses, see Parabola disambiguation. Note as well that a parabola that opens down will always open down and a parabola that opens up will always open up.

We should probably do a quick review of intercepts before going much farther. This means that if we know a point on one side of the parabola we will also know a point on the other side based on the axis of symmetry.

Parabolas can open up, down, left, right, or in some other arbitrary direction. Now, this right over here is an equation of a parabola. Sketching Parabolas Find the vertex. This distance has to be the same as that distance.

The order The parabola here is important.parabola Any point on a parabola is the same distance from the directrix as it is from the focus (F). AC equals CF and BD equals DF. pa·rab·o·la (pə-răb′ə-lə) n. A plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the cone or by the locus of points equidistant from a fixed line and a fixed.

A parabola (plural "parabolas"; Grayp. 45) is the set of all points in the plane equidistant from a given line L (the conic section directrix) and a given point F not on the line (the focus).

The focal parameter (i.e., the distance between the directrix and focus) is therefore given by p=2a, where a is the distance from the vertex to the directrix or focus.

Parabola is a tour de form with force multiplied further. Elegant vision is the constant and ever-changing. And imaginary numbers are the least part of the imagination evident here, and everywhere, in this sublimely sublime book.5/5(4). Aug 01,  · The Music video for Tool's song "Parabola".

In this section we will be graphing parabolas. We introduce the vertex and axis of symmetry for a parabola and give a process for graphing parabolas. We also illustrate how to use completing the square to put the parabola into the form f(x)=a(x-h)^2+k.

Parabola definition, a plane curve formed by the intersection of a right circular cone with a plane parallel to a generator of the cone; the set of points in a plane that are equidistant from a fixed line and a fixed point in the same plane or in a parallel plane. Equation: y2 = 2px or x2 = 2py.

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The parabola
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