Quadratic Equation
ax˛ + bx + c = 0
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One calculates a quadratic equation by multiplying two binomials. This is most easily done by using the FOIL technique. Click here for more information on FOIL and its calculator.
As you most likely have already learned, with every graph (plotted on the coordinate plane) comes an equation. The quadratic equation is used for plotting parabolas. It must be factored out or plugged into the quadratic formula to find out which coordinates on the x-axis the parabola touches.
Luckily, with our calculator you don’t have work at factoring or using the dreaded quadratic formula; just type in the a, b, and c and press solve and the two x-coordinates will be all yours (unless there’s no real solution.) The a, b, and c parts are known as coefficients, and you must know them to use the quadratic formula. For more information on the coefficients of the quadratic equation, click here and go to the second paragraph.
The quadratic equation is used for more than just finding out where the parabola crosses the x-axis, though. It is also used to find both the x and y-coordinates of the parabola’s vertex, the parabola’s axis of symmetry, its y-intercept, and which direction the parabola opens.
The Vertex
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 (like that of the Gateway Arch in St. Louis.)
Coordinates on the Coordinate Plane
One specific point on the coordinate plane is marked in the form: (x, y). This is referred to as an ordered pair. The x- coordinate is the place along the x-axis on which the point is located. The y-coordinate is the place along the y-axis on which the point is located. Obviously, unless the point is in the very center of the coordinate plane—on the point (0, 0)—the point can’t be on both axis’ at once. Therefore, to plot a point given its coordinates, find where the x-coordinate is located along the x-axis and then go up or down from there (depending on whether the y-coordinate is positive or negative) and plot your point where the y-coordinate is. Since the vertex of a parabola is one single point, we use the form of (x, y) to cite it.
X-coordinate
To find the x-coordinate of a parabola’s vertex using the quadratic equation, you start with the equation:
x = -b / 2a
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.
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
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 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. To do 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. They are the two answers you will get by using the quadratic formula or factoring your quadratic equation out.
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
History
The quadratic equation has weathered many transformations from its original form found on clay tablets dating back to 1800 (or so) BC. Some of the names and groups from history most significantly tied to the formation of the quadratic equation are Baudhayana (Indian mathematician,) Babylonian and Chinese mathematicians, Euclid, Brahmagupta, Bhaskara II, and Shridhara.
Example
Find the x-coordinates, vertex, axis of symmetry, and y-intercept of the following quadratic equation:
x² + 5x – 8
and tell which direction it 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:
Do the division:
x = -2.5
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)
Axis of Symmetry
Since the x-coordinate of the vertex is -2.5, we know that the value of the axis of symmetry is also -2.5.
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.
Binomial: an expression consisting of two parts joined by addition or subtraction.
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.
Works Cited
Javed, Syed. “Quadratics.” Math Connections for Adults online. 2005. MCA.
2 June, 2006.
<http://www.staff.vu.edu.au/mcaonline/units/graphs/quads.html>.
“Quadratic Equation.” Wikipedia.com. 5 June 2006. Wikimedia Foundation, Inc. 5 June 2006.
<http://en.wikipedia.org/wiki/Quadratic_equation>.
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