Polynomial Relationship

A verbal description of the relationship

The dependent variable (Y) changes as a the square of the independent variable (X). There may be an offset for X.

The equation and the meaning of x, y, and other parameters

Y = m *(X + b) ^ 2

where b is the offset, the point where Y = 0.

m is a ratio factor between the Y and the (X-b)^2 terms

One or more examples

You are planning to fertilize several square plots with 2 liters of a phosphate solution per meter squared. All of the plots are square but you have to leave a 1.5 meter border around each edge. Calculate the amount of fertilizer you will need for plots of different lengths.

X is the width of the plots

b is (-3) because you have subtract 1.5 meters on both sides of the field

m is 2, the ratio of liters per m^2 of fertilizer

The dimensions of these paramters are very instructive in this equation. X is in meters and thus the offset (b) is also in meters. Y is in liters and thus m has to be in liters per meter^2. For example if you were given two parameters that related to fertilizing the field such as 0.9 liters per m^2 and .7 meters, you would know where to insert those parameters in this equation, because they are only meaningful in one place.

A graph

This graph shows that you would have to apply 0 liters of fertilizer if the plot was only 3 meters across, and that it would increase as a square function as the plot size increases.

If the field was less than 3 meters across, this graph looses its physical interpretation.

 

Algebraic rules that apply to the use of this equation

The most common rearrangement that you might see with polynomials is "factoring" and expanding.

In the example here, (X-3)^2 can be expanded to (X-3)*(X-3) and thus X^2-6X+9.

Sometimes you might encounter the equation in the expanded form and you have to factor it to get back to a simple (X+b)^2 form.

Characteristic values

For the general form of the equation, Y = m *(X + b) ^ 2

at X = 0, Y = m*b^2
at Y = 0, X = -b
Y has a minimum at 0 and increases on either side
X can be any value although negative values may not be meaningful in some cases