Quadratic Functions

Unfortunately the inability to use graphics (i.e. graphs) in writeups currently prevents a more detailed explanation as it stands, however hopefully this will change in a couple of years or so, but as it stands this will have to do. There is also a good chance that your browser will not display some symbols, instead displaying them as various strings of characters. So, for future refrence, ≠ means 'is not equal to'.

The Quadratic Function is any function with a general equation of:

y = ax² + bx + c

Where a, b, and c stand for constants, x is a variable, and a ≠ 0. (Because if a = 0, then the function would be linear and not Quadratic. We call it a Quadratic term because the largest power term is x². If the term with the largest power is higher or lower than to the second, the function is not Quadratic.)

Quadratic functions represent themselves on Cartesian coordinate systems as right side up and upside down "U" like lines. (They are never going from right to center to right or left to center to left because of the fact that functions can only have one y value per x value.) These are called parabolas.

If you know how to plot a graph, you can make your own Parabola using nothing but a pencil, a piece of graph paper and a ruler (Wow!)

1. Start with an equation, for the sake of a common path, lets say y = x² - 4x + 6
2. Make your table of values. For each value of x there is a value of y. I suggest finding from -1 to 6.
3. Plot the values on a piece of graph paper. You should get something that looks like a U. Thats a parabola like we were talking about. Of course if you never learned how to play connect the dots with curves as a kid, it can look a bit messy. :)

That, young seeker of mathematical lore, is a Quadratic Function. As opposed to a Quadratic Equation, (which would be a specific instance) we have discovered a range of possibilities for y based on what x could equal.

While Graphing quadratic functions, some mathematicians realised that there is a lot more to these parabolas than meets the eye.

The Low point of the graph (or the high point of the graph if the co-efficent of the Quadratic Term, x2 is negative: -x2), or more descriptively, the point of the graph where there is no higher or no lower point (that is there can be a higher or a lower point, but not both) is called the Vertex.

Every parabola can be divided into two symmetric parts. This is much more easier to represent with a Quadratic Function that uses a more vertical line of Symmetry. Take for example, y = x² - 6x + 2

If you graph that and then draw the line which divides the parabola into two symmetric parts, you've discovered the line of symmetry! Better known as the Axis of Symmetry.