Parabola
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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.
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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."
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 structures, some of which are parabolas: a roller coaster, the reflector from a flashlight, the base of the Eiffel Tower, the McDonald's arches, and the flight trajectory of NASA's zero-G simulator, known as the "vomit comet".
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.
See Parabolic Bridges for another 'real-world' application.
A More Mathematical Explanation
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
Also, the graph of any equation in this form will be a parabola.
Click below for a derivation of this formula.
This equation provides with information about the curve:
- If then the parabola will open up
- If then the parabola will open down
- The axis of symmetry is given by the line
- The vertex of the curve occurs when . The value of y can be found by substituting x for to the standard form equation.
- In general, when finding the vertex of the parabola.
Vertex Form
Another form to write the equation of the parabola is using vertex form:
In this case, the equation also provides us with important information about the graph:
- As with standard form, if , the parabola will open up
- If , the parabola will open down
- The vertex of the parabola is given by the point
- The axis of symmetry is given by the line
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:
with vertex
- The focus is the point
- The directrix is the equation
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 . Similarly the directrix is also a distance 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
Teaching Materials (2)
Want to add your own teaching materials or lesson plans? Add teaching materials.
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
- Clean up the section on real world parabolas.
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