Parabola

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Revision as of 14:07, 29 July 2011

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.

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 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.

Real World Parabolas

The appearances of parabolic shapes in the physical world are very abundant. Here are a few examples with which you may be familiar. Below you will see five real-world parabolic structures: a roller coaster, a beam of light from a flashlight, the base of the Eiffel Tower, the McDonald's arches, and a plane's flight trajectory.

Have you ever wondered why the best roller coasters are parabolic? When you're riding these coasters it feels like you're defeating the force of gravity, right? Exactly! When a coaster falls from the peak of the parabola, it is rejecting air resistance and all the bodies are falling at the same rate. The only force here is gravity.

G forces also play a role in your feeling of weightlessness and heaviness when falling. First, it is important to distinguish that gravity is not a G force. G force is measured in what you feel while sitting still in the earth's gravitational field. So, for example, when you are seated in a coaster, the seat is exerting the same amount of force as the earth, but it is directed oppositely, this is what keeps you sitting still. But, when you're falling you aren't experiencing any G's (zero G force) so actually the seat isn't supporting you at all! But once you are at the bottom of a drop, the G force returns to being greater than 1:

• G forces are greatest at the minimums of a parabola
• G forces are least at the maximum of a parabola

So, the shape of a coaster as well as the ascent and descent play a vital role in the rider's enjoyment. Parabolic-shaped coasters are enjoyed so much because of the intense pull of gravity and the nonexistent G force that occurs when falling.

A More Mathematical Explanation

Figure 3
Formally, a parabola is the set of all points [...]

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).

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References

• A derivation of the value $\tfrac{1}{4a}$
• Clean up the section on real world parabolas.