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 with a line and a point off to one side. The line is called the "directrix"; the point is called the "focus". The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola up the middle) is called the "axis of symmetry". The point on this axis which is exactly midway between the focus and the directrix is the "vertex"; the vertex is the point where the parabola changes direction.

"regular", or vertical, parabola (in blue), with the focus (in green) "inside" the parabola, the directrix (in purple) below the graph, the axis of symmetry (in red) passing through the focus and perpendicular to the directrix, and the vertex (in orange) on the graph

"sideways", or horizontal, parabola (in blue), with the focus (in green) "inside" the parabola, the directrix (in purple) to the left of the graph, the axis of symmetry (in red) passing through the focus and perpendicular to the directrix, and the vertex (in orange) on the graph

The name "parabola" is derived from a New Latin term that means something similar to "compare" or "balance", and refers to the fact that the distance from the parabola to the focus is always equal to (that is, is always in balance with) the distance from the parabola to the directrix. In practical terms, you'll probably only need to know that the vertex is exactly...

...are four types of conic sections, circles, parabolas, ellipses, and hyperbolas. The first type of conic, and easiest to spot and solve, is the circle. The standard form for the circle is (x-h)^2 + (y-k)^2 = r^2. The x-axis and y-axis radius are the same, which makes sense because it is a circle, and from
In order to graph an ellipse in standard form, the center is first plotted (the (h, k)). Then, the x-radius is plotted on both sides of the center, and the y-radius is plotted both up and down. Finally, you connect the dots in an oval shape. Finally, the foci can be calculated in an ellipse. The foci is found in the following formula, a^2 b^2 = c^2. A is the radius of the major axis and b is the radius of the minor axis. Once this is found, plot the points along the major axis starting from the center and counting c amount both directions.
In order to determine if an equation is an ellipse, the following three criteria must be met. There must be an x^2 and a y^2 just like in a circle. However, the coefficients of the x^2 and y^2 must be different. Finally, the signs must be the same. For example, equation 4 is an ellipse. 49x^2 + 25y^2 +294x 50y 759 = 0 has an x^2 and a y^2. It also has different coefficients in front of them, and finally, both have the same sign! There you have it, an ellipse!HyperbolasBoy, now it is starting to get tough! But dont worry, hyperbolas are not much more difficult than ellipses. Imagine two parabolas opposite...

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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...

...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: ZA: BG: A B. AG. This line ZT is called the latus rectum; it is now also called the parameter. Now take any point whatever, as K, on the curve. From it draw KL parallel to DE meeting the diameter in L. ZL is called the abscissa. If now, on ZL as a base, we erect a rectangle equal in area to the square on KL, the other side of this rectangle may be precisely superposed upon the latus rectum, ZT. This property constitutes the best practical definition of the parabola. If a similar construction were made in the case of the ellipse, the side of the rectangle would fall short of the latus rectum; in the case of the hyperbola, would surpass it. The modern scientific definition of the parabola is that it is that plane curve of the second order which is tangent to the line at infinity. The parabola is also frequently defined as the curve which is everywhere equally distant from a fixed point called its focus, and from a fixed line called its directrix. The normal to a parabola at every...

...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 both the directrix and the focus is the parabola. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a planewhich is parallel to another plane which is tangential to the conical surface.[a] A third description is algebraic. A parabola is a graph of a quadratic function, such as
The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola which is parallel to the directrix and...

...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 at it you can see the apparent parabolic curve in his jumps and how it changes in as he changes. As time countless games used parabolic curves in there video games, so many that it probably would be impossible to count. Angry Birds is another game that uses parabolic curves as its main use of action. The uniqueness of Angry Birds though is the ability to change the curve to almost any form that is useful to the player to accomplish any goal that is set in front of them. This game has proven to be popular and has received many sequels just like Mario did. It is apparent that the combination of these two things together, video games and parabolas, become very popular and sell very well. Even one of the most popular gaming franchises now, Call of Duty, uses parabolas in many of its gameplay from mortars to grenades. Even in decorations for maps like rainbows and banners you can see the curve apparent. The need for parabolas in video games is to make them realistic which is what people always want....

...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 theparabola widens due to the fact that (a) is essentially the slope of the parabola. Now, watch what happens when the (a) becomes negative.
Now the mouth of the parabola is opening down and as we increase (a), the parabola widens. When (a)1, the parabola closes up horizontally. So from this, we can infer that if (a) is 0, then the result will be a straight line going along the X axis which is no longer a parabola. Now we will look at how the (b) factor changes the parabola, below is the origional graph from the top of the paper except the red line represents the same parabola with a (b) added onto it. The original equation was Y=x^2(blue line) and the new equation is Y=x^2+x(red line). The change is moving the vertex of the parabola left ½ and down ¼ but the shape of the parabola it’s self is unchanged. Here are some other examples of changing (b) in...

...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 theparabola is used throughout the world of structures. Most of the time, it is used as an arch or an arc. Have you ever seen a castle or even a movie with a castle in it? Well usually on the big great front doors, you will see it lined with stones ending in a curve at the top. Usually the top-center stone is the biggest. It is the key stone to an arc structure like that. All the other stones in the curve try and slide into the middle, but they are stopped by the one on top. They all squeeze together towards the middle and the key stone at the center takes all this pressure and keeps it together. If something is set on top of that curve, the curve will take the weight and try and push towards the middle, but since the stones are set to a curve, they push on each other and stay firm. This design is used in bridges and even cars for strength.
Another use of the parabola is in lights. Headlights, searchlights, flashlights and more. The reflective mirrors inside are one big...

...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
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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 when we reached the high point, an 18,000-foot pass over a crest that we'd have to traverse to reach the village of Muklinath, an ancient holy place for pilgrims.
Six years earlier, I had suffered pulmonary edema, an acute form of altitude sickness, at 16,500 feet in the vicinity of Everest base camp-so we were understandably concerned about what would happen at 18,000 feet. Moreover, the Himalayas were having their wettest spring in 20 years, hip-deep powder and ice had already driven us off one ridge. If we failed to cross the pass, I feared that the last half of our once-in-a-lifetime trip would be ruined.
The night before we would try the pass, we camped in a hut at 14,500 feet. In the photos taken at that camp, my face appears wan. The last village we'd passed through was a sturdy two-day walk below us, and I was tired.
During the late afternoon, four backpackers from New Zealand joined us, and we spent most of the night awake, anticipating the climb. Below, we could see...