Quadratic Functions and Their Graphs
Several Examples of quadratic functions are in the table below.
| $$x^2+2x+1$$ Fairly standard quadratic function with integer coefficients. | $$x^2-4$$ In this case, \(b=0\) | $$5x^2$$ In this case, \(b\) and \(c\) are both \(0\). | $$3.14x^2-5.123x+9$$ Nothing wrong with decimal coefficients! |
| $$\frac{1}{2}x^2-x+\frac{3}{5}$$ Nothing wrong with fractions either! | $$-x^2-7x-9$$ Can certainly have negative coefficients! | $$-51.2x^2+6.42x-70.48$$ Most real-world quadratic functions look like this. | $$-82.513x^2$$ Not too bad-looking. |
Notice that in all the above examples, the one thing that they all have in common is they all have an \(ax^2\) term for some \(a\). Quadratic functions do NOT need to have an \(bx\) term or a \(c\) term (in these cases, both \(b\) and \(c\) are negative).
Graphs of Quadratic Functions
Quadratic functions have the shape of a parabola, which is more or less the shapes you see in the examples below. Some quadratic functions have wider parabolas and some have narrower ones. Some even are “upside down,” compared to the usual “cup-shaped” parabola.
Using Quadratics To Model Data and More Complicated Graphs
As mentioned before, many real-world phenomena can be modeled very well using parabolas. For instance, lets look at the price chart of a particular asset over the course of years:
The graph not only indicates an increasing price over time, but an increasingly increasing price over time; i.e. the rate of increase is itself increasing. When this sort of thing happens, we could use a smooth curve like that of (a portion of) a parabola to model an “average price” over time. A parabola could also be chosen to “fit the data” in an optimal way, so that the points on the jagged function’s graph are all no more than a certain distance away from the parabola. One example of a parabola that models the data in the price graph is given below.
There are ways of finding the formula for the parabola that best fits a given data set or graph such as the one above, but the methods for doing so will not be explored here. For now, just note that you can indeed find one either by hand or mechanically using various graphing or spreadsheet applications.