All About Those Funky Parabolas
Related Calculators |
Additional Information |
Baseball, fountains, the Golden Gate Bridge, arched doorways in cathedrals. What do all of these have in common? Answer: they all have to do with parabolas.
A parabola is the arch-like graph of a quadratic equation. It opens up or down (depending on what your quadratic equation is) to make the shape of a necklace hanging down from someone's neck or the Gateway Arch in St. Louis. Once you are given a parabola's quadratic equation, you can be on your way to graphing it on a coordinate plane.
With the a, b, and c coefficients of the equation you start with and a few other key equations, you can find the parabola's x-coordinates , vertex , axis of symmetry , y-intercept, and direction it opens. If you're not sure what I mean by a, b, and c, coefficients, hang tight; we'll get to that in just a minute.
I will now go on to explain each of the terms above that are in italics. After all that, I'll explain what the term "discriminant" on the grapher refers to.
X-coordinates (referred to as "neg zero pt." and "pos zero pt." on the grapher)
A parabola's x-coordinates are the points that its two arms cross the x-axis.You can find these points by either using the quadratic formula or factoring your equation out.
The terms "neg zero pt." and "pos zero pt." are used in our grapher because at the ending stage of the quadratic formula, you must add and subtract two values together to get your two answers for x.
The Vertex (referred to as "Coord." on the grapher)
The parabola's vertex is the point on the coordinate plane the parabola starts from. This is the point from which the arms of the parabola extend outwards to make either the shape of a necklace or the shape of an arch.
X-coordinate(referred to as "x-top" on the grapher)
To find the x-coordinate of a parabola's vertex using the quadratic equation, you start with the equation:
If you're wondering where to get the b and a values from, simply look at the quadratic equation you started with.
Use whatever number is in front of the x² as a (if no visible number is in front of it, your a value is 1) and whatever is in front of x as b. Also (for future reference,) use the number with no x value next to it to get the value of c (if you don't see any number with no x next to it in your equation, use the number zero for c.)
This will give you the x-coordinate of the vertex. With that value, you can now find the y-coordinate of the vertex.
Y-coordinate (referred to as "y-top" on the grapher)
To find the y-coordinate of the vertex, you replace the x variable on the quadratic equation you started with with your value for the x-coordinate. So, if by doing the equation above, I found my x-coordinate to be 3, I would write my equation as:
a(3)² + b(3) + c
With your two values for the x and y-coordinates, you now have the point of the vertex.
The vertex is referred to as "coord." (an abbreviation for "coordinates") on our grapher because the vertex is the coordinates at which the parabola starts from.
The Axis of Symmetry
The axis of symmetry of a parabola is pretty much a fancy-sounding term for the x-coordinate of its vertex. The main point is that if you were to draw a line directly through the center of the parabola, you'd have its axis of symmetry. Since this line would be in the exact center of the parabola, whatever was to the left side and the right side of it would be the exact same and have the same numeric values. Therefore, each side of the parabola would be symmetrical.
To calculate the value of the axis of symmetry of a parabola, take the x-coordinate of its vertex. That is your answer.
In case you don't have (or aren't required to) find the vertex of the parabola but do need to find the axis of symmetry, you must find the points where the parabola crosses the x-axis. After doing this, simply divide the sum of the value of these points by 2. The equation for this process is:
(x1 + x2) / 2
x1 and x2 in this equation simply stand for the two places where the parabola crosses the x axis (referred to as "neg zero pt." and "pos zero pt." on our grapher.
They are the two answers you will get by using the quadratic formula or factoring your quadratic equation out. Both of these processes for finding x-coordinates are explained in our Quadratic Equation and Factoring help files. If your parabola doesn't cross the x-axis then you'll just have to find the x-coordinate of its vertex and get your answer that way.
The Y-intercept
The y-intercept of a parabola refers to the point at which one of the parabola's arms hits the y-axis. One might want to know this if he/she wishes to make his/her graph of the parabola as accurate as possible (if he/she must draw it by hand.)
For example, using the y-intercept and the axis of symmetry, you can easily calculate how wide the parabola opens. Since each side of the parabola is equal distance from the axis of symmetry, however far apart the y-intercept is from the axis of symmetry will be as far apart as the other side is from the axis of symmetry.
The way to formulate this intercept is to replace every x variable in your quadratic equation with a zero. So, your equation would look like this:
a(0)² + b(0) + c
You may realize that every time you calculate a quadratic equation (that is in the standard format) to find the y-intercept of its parabola, your answer will always be whatever you have as your c value.
If your equation doesn't have a c value (the c value is zero) that just means that your parabola crosses the y-axis at the center of the coordinate plane (0, 0) and, therefore, one of your values for x will be zero also.
Direction that the Parabola Opens
It is simple to find out whether a parabola opens up or down. All you do is check to see if the ax² in your quadratic equation is positive or negative.
The parabola of this equation will open upwards (and will look like a necklace):
x² + 6x - 4
The parabola of this equation will open downwards (and will look something like the Gateway Arch in St. Louis):
-x² - 6x + 4
Example
Find the x-coordinates, vertex, axis of symmetry, and y-intercept of the parabola that the following quadratic equation graphs:
x² + 5x - 8
and tell which direction the parabola opens.
X-coordinates
Since this equation looks like it would pretty hard to factor out, let's use the quadratic formula to find its x-coordinates:
Vertex
First, let's find the x-value of the vertex:
With our value of the x-coordinate, we can now calculate the y-coordinate.
The first thing we do is replace each x in our original quadratic equation with the value we just got for the x-coordinate:
y = (2.5)² + 5(-2.5) - 8
Now we do the squaring and multiplication:
y = 6.25 -12.5 - 8
Now solve starting from the left of the equation (click here to find out why you start from the left):
y = -14.25
Therefore, our vertex is:
(-2.5, -14.25)
Y-intercept
To find the y-intercept of the parabola, replace each x in our original quadratic equation with zero:
(0)² + 5(0) -8
From there, our calculation is quite easy:
0 + 0 -8 = 0 - 8 = -8
Our parabola will cross the y-axis at -8.
Direction the Parabola Opens
Since the ax² value of our quadratic equation is positive (doesn't have a negative sign in front of it,) our parabola will open upwards and, therefore, will look something like a necklace hanging from a person's neck.
Discriminant
A discriminant is not actually a part of a parabola. It is a term that refers to one part of the quadratic formula which is used to graph parabolas. Click the link to find out all about the quadratic formula and how to use it.
The discriminant of your quadratic formula tells you whether your parabola will cross the x-axis in two places, one place, or nowhere at all.
If your discriminate is greater than zero, your parabola will cross the x-axis in two places; if it's equal to zero it will cross the x-axis in one place; and if it's less than zero, you will get no real solution meaning your parabola won't cross the x-axis at all.
History
The Greek mathematician and scientist Apollonius gave the parabola its name. The word most likely originated from the Greek word paraboleuvomai which, according to Foreignword.com, translates to English as "a placing of one thing by the side of another" or "a comparison of one thing with another; likeness." This makes sense if you think of a parabola according to its line of symmetry...each side of every parabola is placed next to and very similar to the other.
It is thought that the Greek mathematician Menaechmus/Menaichmos discovered the parabola while trying to double the area of a cube. He used the intersection of two parabolas to find an answer to this problem.
Soon after, Galileo studied the laws of gravity to discover that certain projectiles fall in parabolic paths.
Helpful Definitions
Coefficient: an unknown value preceding a known one, joined to it through multiplication.
Coordinate: a number corresponding to a certain location on a coordinate plane.
Vertex (of a parabola): the point from which the arms of a parabola ascend or descend.
Axis of symmetry (of a parabola): an imaginary line that divides a parabola so that each side reflects the other.
Symmetrical: having each side share the same shape, size, and position of the other.
Related Calculators |
Additional Information |
Works Cited
Weisstein, Eric. "Parabola." MathWorld. 1999. Wolfram Research, Inc. 6 June 2006.
< http://mathworld.wolfram.com/Parabola.html>.
"The New Testament Greek Lexicon." foreignword.com. 1995. Crosswalk.com. 6 June
2006. <http://www.foreignword.com/Tools/dictsrch_aff.asp?menu=N&query=parabola&src=CQ&trg=BP>.
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
