Parabola

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Parabola
Field: Geometry
Image Created By: Unkown
Website: Absolute Ponds

Parabola

A parabola is a u-shaped curve that arises not only in the field of mathematics, but also in many other fields such as physics and engineering.


Contents

Basic Description

Parabolas are a common shape: for example, a stream of water from a hose or fountain, starting upward, curving as it nears the peak, and straightening out somewhat as it heads back down. It's the path followed by any thrown object, but it's easiest to see with water. The path is called a "parabolic trajectory."



Figure 1
Figure 1
Figure 2
Figure 2
Another way to describe this curve is using a cone. Imagine you have an ice cream cone and slice it so that it is cut parallel to the slope of the cone. The new edge formed is in the shape of a parabola.
The blue line in figure 1 is the curve. Figure 2 shows the "slice" or the conic section within the cone that becomes the parabola.






A More Mathematical Explanation

Figure 3
Figure 3
Formally, a parabola is the set of all points in the plane equidistant from a line and a given point not on the line. In figure 3, the red dot F is the focus and the dashed line L is the directrix. The line segments \overline{FP_1} and \overline{P_1Q_1} are of equal length. Thus, the point P_1 is equidistant from the focus and the directrix. The same occurs with points P_2 and P_3. By drawing infinitely many points P such that the distance from F and from a point Q in L are equivalent, a parabola is formed.












Equation of the Parabola

Graphs of parabolas can be oriented in any direction: upwards, downwards, sideways, even diagonally. The following equations and examples for the parabola will be written oriented vertically to maintain consistency.

Standard Form

Any vertically oriented parabola can be written using the equation 

y=Ax^2+Bx+C

Also, the graph of any equation in this form will be a parabola.
Click below for a derivation of this formula.

Derivation

Let the point (a,b) be the focus, y=d be the directrix, and (x,y) be any point on the parabola.

  • By definition, the distance from the focus to the point (x,y) is equal to the distance from the directrix to (x,y).

D=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}


  • We then substitute the values of the focus (a,b) and the point (x,y) into the distance equation.

D_f=\sqrt{{(a-x)}^2+{(b-y)}^2}


  • We do the same with the distance from the corresponding point on the directrix (x,d) to (x,y).

D_d=\sqrt{{(x-x)}^2+{(d-y)}^2}

  • Apply the definition of Parabola that states that the distance from a point to the focus is equal to the distance from that point to a point on the directrix.

D_f=D_d, so

\sqrt{{(a-x)}^2+{(b-y)}^2}=\sqrt{{(x-x)}^2+{(d-y)}^2}

  • We simplify and square both sides.

{(a-x)}^2+{(b-y)}^2={(d-y)}^2


  • We collect and expand all the y terms on the same side.

{(a-x)}^2={(d-y)}^2-{(b-y)}^2

{(a-x)}^2=d^2-2dy+y^2-(b^2-2by+y^2)


  • We then cancel and combine like terms and expand the left side.

x^2-2ax+a^2=d^2-b^2+2(b-d)y


  • Now, we solve for y.

x^2-2ax+a^2-d^2+b^2=2(b-d)y
and

\frac{x^2}{2(b-d)}-\frac{2ax}{2(b-d)}+\frac{a^2}{2(b-d)}+\frac{b^2-d^2}{2(b-d)}=y


  • We can now simplify.

\frac{1}{2(b-d)}x^2+\frac{a}{(d-b)}x+\frac{a^2}{2(b-d)}+\frac{b+d}{2}=y


  • Since a,b, and d are all real numbers, we can redefine some values.

A=\frac{1}{2(b-d)}
B=\frac{a}{(d-b)}


C=\frac{a^2}{2(b-d)}+\frac{(b+d)}{2}

  • We substitute in these values.

Ax^2+Bx+C=y


This equation provides with information about the curve:

  • If a>0 then the parabola will open up
  • If a<0 then the parabola will open down
  • The axis of symmetry is given by the line x=\frac{-b}{2a}
  • The vertex of the curve occurs when x=\frac{-b}{2a}. The value of y can be found by substituting x for \frac{-b}{2a} to the standard form equation.
  • In general, y=\frac{-b^2}{4a}+c when finding the vertex of the parabola.



Vertex Form

Another form to write the equation of the parabola is using vertex form: 

y=a{(x-h)}^2+k


In this case, the equation also provides us with important information about the graph:

  • As with standard form, if a>0, the parabola will open up
  • If a<0, the parabola will open down
  • The vertex of the parabola is given by the point (h,k)
  • The axis of symmetry is given by the line x=h


It is possible to convert one form into the other. To go from vertex to standard form, we can just simplify the equation by squaring the parentheses and combining like terms. Sometimes there are short cuts to go from standard to vertex form, but in general, it will require that we complete the square.

In both forms, to find the x-intercepts we let y be zero and we solve for x. Conversely to find y-intercepts we let x be zero and solve for y.

Finding the Focus and Directrix

Sometimes, it may be useful to determine the values of the focus and of the directrix. The focus is located inside of the curve a distance P from the vertex of the parabola. The directrix is a line located at a distance P from the vertex in the opposite direction. For the general equation of a parabola in standard form:
y=ax^2+bx+c with vertex \left ( \frac{-b}{2a}, \frac{-b^2}{4a}+c \right )

  • The focus is the point \left ( \frac{-b}{2a}, \frac{-b^2}{4a}+c+\frac{1}{4a} \right )
  • The directrix is the equation y=\frac{-b^2}{4a}+c-\frac{1}{4a}

Notice the relation between the values of the vertex, the focus, and the directrix. The difference between the y-values of the vertex and the focus is \tfrac{1}{4a}. Similarly the directrix is also a distance \tfrac{1}{4a} but in the opposite direction, thus subtracted from the Y value of the vertex rather than added (like the focus).

Interesting Applications of the Concept







References

http://commons.wikimedia.org/wiki/File:Conicas2.PNG

http://en.wikipedia.org/wiki/Parabola

http://mathworld.wolfram.com/Parabola.html

http://mathforum.org/library/drmath/view/54390.html

Future Directions for this Page

  • A derivation of the value \tfrac{1}{4a}


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