Top-Rated Free Essay

# The Parabola

Good Essays
The Parabola
What is a Parabola?
A quadratic expression is an expression in which the highest power of is 2. Consider the following: The above equations are all quadratic expressions as the highest power of is 2. When a quadratic function is graphed, the resulting curve is called a parabola, as demonstrated in figures 1 and 2.

Real Life Application
A quadratic function and parabola can be used when undertaking a new business venture to determine the optimum sales price of a particular new product and therefore predict unit sales, sales in dollars, costs and profit.
Take for example an entrepreneur starting up a music store that both produces and sells brass instruments. The owner, Joe, has designed a new professional grade euphonium and wants to mass produce them and sell them online for maximum profit. Joe has estimated that his costs are going to be:
\$900,000 for manufacturing set-up, promotion and advertising costs
\$3,000 to make each euphonium
Joe has also undertaken extensive research into his market and based on his competitors, expects sales to follow a “Demand Curve” similar to: Unit Sales = 350,000 - 80P; where P equals price. If Joe sets the price at \$0, he is giving away all the Euphoniums for free, or, by selling them at \$4000, he will sell 30,000 euphoniums. However, the question still remains, what is the best price for Joe to sell his euphoniums and how much profit should he make? This is where the quadratic function and the parabola comes in.
Joe must first come up with a quadratic expression. How much he sells depends on the price of the product. Experimenting with the equation, we use price (P) as the variable:
Unit Sales =
Sales in Dollars = Units x Price =
Costs
Profit = Sales – Costs = (A quadratic equation)

We now have a quadratic equation that can be solved by completing the square.

(nearest whole number)

The above equation shows that Joe will earn no profit when the price is \$4366.50 or \$3008.50. This means that maximum profit will be exactly halfway between, shown on our parabola:
Where the maximum point of the parabola is, we have maximum profit directly in line with the optimal sale price. Therefore, as shown in figure 3, the ideal sales price is \$3687.50. This means:
Unit Sales =
Sales in Dollars = Units x Price =
Costs
Profit = Sales – Costs =

Through the use of a parabola and quadratic expression, Joe has been able to determine the optimum sales price for his product, \$3687.50, and estimate that it will be quite a profitable business venture, earning a total profit of \$36,912,500. The use of parabola for predicting sales can therefore be useful not only for new businesses or entrepreneurs to determine if a business venture will be profitable, but for existing businesses looking for renewal or growth in the development of new products.
Domain and Range Issues
For any parabola, we can find the domain and range where:
Domain: set of all possible x – values
Range: set of all possible y – values
There are a number of restrictions on the domain:
Cannot divide by zero
Cannot take a negative root
For the above parabola used in the above example, we can change the values of p to represent x, therefore,, and can find the domain and range where:
Domain
None of the above restrictions apply. Any real value of x can be used.
Range
By interpreting the graph, we can see the maximum value of y = . Therefore, as this is the maximum value, all other values of y must be less than, alternatively, .
Therefore:
Domain: all real x
Range:
As shown above, there are no real domain and range issues that affect the parabola , as there are no real values of x that cannot be used, and the corresponding y - value is a direct result of these.

Bibliography
Internet Articles
1. "Sketching Parabolas." Maths Online”. N.p., n.d. Web. 11 June 2014. Available: http://www.mathsonline.com.au/students/parabola
2. "Real World Examples of Quadratic Equations." Real World Examples of Quadratic Equations. N.p., n.d. Web. 11 June 2014. Available: http://www.mathsisfun.com/algebra/quadratic-equation-real-world.html.
Figures
1. "Sketching Parabolas." Maths Online”. N.p., n.d. Web. 11 June 2014. Available: http://www.mathsonline.com.au/students/parabola
2. "Sketching Parabolas." Maths Online”. N.p., n.d. Web. 11 June 2014. Available: http://www.mathsonline.com.au/students/parabola
3. Sketch, Connor Hutchinson

## You May Also Find These Documents Helpful

• Good Essays

Conics: Parabolas: Introduction (page 1 of 4) Sections: Introduction, Finding information from the equation, Finding the equation from information, Word problems & Calculators In algebra, dealing with parabolas usually means graphing quadratics or finding the max/min points (that is, the vertices) of parabolas for quadratic word problems. In the context of conics, however, there are some additional considerations. To form a parabola according to ancient Greek definitions, you would start…

• 1038 Words
• 5 Pages
Good Essays
• Satisfactory Essays

Theory of Parabolas By Amergin McDavid A parabola is designed on a basic formula, Y=ax^2+bx+c, which allows it to achieve a curve not seen in a normal line graphed using a Y=mx+b format. To the left is a graph who’s formula is y=x^2, where a=1, b=0, and c=0. I have isolated the (a) factor to see its effects on the parabola. Below is a graph where I have changed the (a) multiple times. The result is that as the (a) decreases, the mouth of the parabola widens due to the fact that (a) is essentially…

• 373 Words
• 2 Pages
Satisfactory Essays
• Satisfactory Essays

A parabola is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram, but which can be in any orientation in its plane. It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The locus of points in that plane that are equidistant from…

• 365 Words
• 2 Pages
Satisfactory Essays
• Good Essays

The Importance of the Parabola What exactly is a parabola? Well it could quite possibly be the most powerful shape that our world has ever known. It is used in many designs since it is so sturdy and powerful. Countless structures and devices use the parabola and it does nothing but enhance whatever it is used in. What makes it so powerful? Just keep reading and find out. Used in bridges, doors and buildings, the shape of the parabola is used throughout the world of structures. Most of the time…

• 461 Words
• 2 Pages
Good Essays
• Good Essays

My parabola is the arc of of which a diver’s body creates when he is in the pike position. It is irrelevant whether he is on the board, in the air, or on the ground. As long as the person is in the position that it all that matters. This parabola can be created anywhere. As long as you can get into a pike position, the location does not really matter. However, a diving board helps the diver to jump up higher to create a more distinguished parabola. Diving can be found in most first world countries…

• 598 Words
• 3 Pages
Good Essays
• Good Essays

DE be a line perpendicular to this triangle, cutting BG in H. Let the cone be cut by a plane through DE parallel to AG, so that the intersection with the cone will be the curve called the parabola. Let Z be the point where this curve cuts AB. Then the line ZH is called by Apollonius the diameter of the parabola, or the principal diameter, or the diameter from generation; it is now called the axis. From Z draw ZT at right angles to ZH and in the plane of ZH and AB, of such a length as to make ZT:…

• 649 Words
• 3 Pages
Good Essays
• Good Essays

circumference consists of points iuuuuequal distance from the center Parabola: a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side Hyperbola: a symmetrical open curve formed by the intersection of a circular cone with a plane at a smaller angle with its axis thanDefintions Circle: a round plane figure that the circumference consists of points iuuuuequal distance from the center Parabola: a symmetrical open plane curve formed by the intersection of…

• 3660 Words
• 15 Pages
Good Essays
• Satisfactory Essays

Goodman Pre-AP Algebra 2 Coach Crowder February 22, 2013 Analysis When thinking of video games you don’t think about math at all. The two things just seem so far apart that they can’t be together. In all reality video games are filled with math. Parabolas are more common in video games than most people really notice. As mentioned before Mario is one of the most popular games ever to be created. When created it looked like a simple game about a plumber trying to save a princess, but when looking closer…

• 278 Words
• 2 Pages
Satisfactory Essays
• Good Essays

The Parable of the Sadhu The following case first appeared in the September-October 1983 issue of the Harvard Business Review. It was written by business professor Bowen H. McCoy and is a true story ---------------------------------------- The Nepal experience was more rugged than I had anticipated. Most commercial treks last two or three weeks and cover a quarter of the distance we travel. My friend Stephan, the anthropologist, and I were halfway through the 60-day himalayan part of the trip…

• 1683 Words
• 7 Pages
Good Essays
• Satisfactory Essays

A parabola can easily be viewed as an elegant arched trajectory naturally formed by any projectile, from an artillery round to a tomato, moving in a gravitational field. Parabolas have been extensively studied since people started throwing stuff at each other, and they shape the outcome of many ballistic sports, such as baseball, golf, football, shot put and more. But they reach their apex in basketball, where field goals and free throws demand precision control of parabolas. So in tribute to the…

• 194 Words
• 1 Page
Satisfactory Essays