Parabola - The Arch Enemy?

I love to relate math to the real world with my students because that is the only way they will see the relevance.  Our family (son + his wife + three grandkids + husband) just returned from a trip to Orlando.  While driving home from Cocoa Beach, my daughter-in-law noticed a purple arch.  Being that my mind is always, always thinking about math, I informed her that it was a parabola with a negative slope.  My son, who is an engineer, starting talking about slope, and the two of us shared some equations such as the one for lines y = mx + b (much to the chagrin of the other travelers stuck in the car with us).

Mathematically speaking, a parabola is a two-dimensional, symmetrical curve or simply, a special curve shaped like an arch.  All parabolas are vaguely “U” shaped, and they have a highest or lowest point called the vertex.  The vertex is the place the parabola makes it sharpest turn.  Any point on a parabola is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix). Parabolas may open up or down and may or may not have x-intercepts, but they will always have a single y-intercept. Those that open up have a positive slope (they form a smile) and those that open down have a negative slope (they form a frown).  We always get a parabola when we graph a quadratic equation, an equation that contains a variable that is squared such as y² = 20x or x² - 9y = 0.

Now that all of this is as clear as mud for many of you, let's look at some parabolas in real life.  Yes, they are out there!  Can you identify the ones below?

Did the last picture stump you?  Well, it would unless you were from Los Angeles.  It is the

Encounter Restaurant, atop Los Angeles International Airport’s landmark Theme Building.

Other real life examples include....

1) Throwing or Kicking a Ball - If you throw a baseball, kick a soccer ball, shoot an arrow, fire a missile, or throw a stone, it will arc up into the air and come down again following the path of a parabola! (Except for how the air affects it.) The next time you watch a football being thrown from the quarterback to a receiver, think of a parabola.

2) Roller Coasters that arc up and down and sometimes around - the one ride I avoid! When a coaster falls from the peak (vertex) of the parabola, it is rejecting air resistance, and all the bodies are falling at the same rate. The only force here is gravity. Most people (I am NOT included) enjoy or get a thrill out of parabolic-shaped coasters because of the intense pull of gravity.

3) Reflectors - Parabolas are also used in satellite dishes and flashlights. In satellite dishes it helps reflect signals that then go to a receiver, which interprets the signals and shows satellite-transmitted channels on your television. In flashlights, car headlights and spotlights, the parabolic shape helps reflect light. Notice the beam of light coming from the flashlight on your right. See how the light appears to be in the shape of a parabola?

4) Suspension Bridges such as the Golden Gate Bridge, the Brooklyn Bride, the Washington Bridge, etc.  Suspension bridges are capable of spanning long distances and actually are the only type of bridge to span the longest distances possible for a bridge. This is because the shape of the suspension bridge is actually one of the most stable structures there is. In the image of above, can you see how the cables form parabolas?

So now you know parabolas are everywhere even when you are playing ports, watching T.V., riding a roller coaster at your favorite theme park or going cross a suspension bridge.  So what kind of parabola will you display on your face today…a negative parabola (a frown?) or a positive parabola (a smile)?