Probably the simplest, but not necessarily the "neatest", nonlinear function is the quadratic (or parabola). This is a curve whose slope is changing at a constant rate. One of the implications of this is that eventually it will level out and reverse directions. It is useful to describe phenomenon that reach a minimum or a maximum at some point.

The classic example of a parabola is simple projectile motion, where an object tossed in the air is slowed down and eventually reversed by the constant acceleration of gravity. From a business perspective we tend to see a quadratic when the sales price is a linearly decreasing function (as indicated by the Law of Demand) of production (or sales volume). This means that while increasing sales initially has a beneficial effect on revenue (which is sales price times quantity), eventually the price will drop so low that the amount of sales can't compensate and revenue will drop as well. The equation describing this revenue (and ultimately your profit as well) would then be a quadratic function.

While there are a lot of nifty aspects of parabolas to study, we are mostly concerned with identifying certain useful features - namely the y-intercept, the x-intercepts, and the vertex. Not only do these tend to be critical points of whatever phenomena we are studying, but they enable us to draw a rough sketch of the parabola (without the assistance of a graphing device). Here are some mystery functions to find equations for.

Finding x-intercepts

Finding where y = ax²+bx+c = 0 is a familiar problem. In certain cases it can be solved by factoring the trinomial and using the zero product rule to set each of the factors equal to zero and solve for x. This assumes that there is a nice whole number solution. Otherwise we have to resort to the quadratic formula for x:

and 

Finding the vertex

While both the x-intercepts and the vertex can be approximated graphically, the vertex also has a handy formula. It actually can be observed from the quadratic formula that the point betwee the two x-intercepts is h = -b/2a. This is the horizontal coordinate of the vertex. This value can be plugged back into the quadratic function to determine the vertical coordinate, but the result can be further simplified as k = bh/2+c.

Various Graphing Spreadsheets involving Quadratic Functions

Various functions - Enter the scale factor, the horizontal and vertical shift of the parabola and it will generate the corresponding curve. Includes several other basic functions as well, so make sure you select the "Quadratic" tab at the bottom.

Intersection of two curves - Just type in two quadratic equations (or one linear and one quadratic) in the indicated locations and click on the buttons (involves Excel macros), and it will automatically generate the data for the two curves and force the last point to be the intersection of them. It will only find one of the two intersections (whichever is closest to the original last point). To find an x-intercept, make one of the two equations y=0.

Minimum or Maximum of a parabola - Enter the Excel calculation for the first y value of the parabola and use autofill to generate the remaining ones. When you click on the button it will rearrange the points so that the second to last point is the minimum or maximum of the parabola.

Contact Leo Wibberly at ldwibber@vcu.edu or (804) 740-4650 to make appointments