Graphing and Mathematical Models

Section 2.7: Graphing and Mathematical Models

In the previous section, we turned the various graphing principles that we have observed in previous sections into a step-by-step graphing strategy, and applied that technique to explain various algebraic formulas. We also showed how to reverse the process to determine the equation of a linear function, deriving the point-slope formula. In this section, we will show how to extend this idea to discover the mathematical formula corresponding to a graph derived from a larger class of "core" functions. As we said in the Introduction, the ability to "fit" a formula to data is foundational to the scientific method, since it is how we can move from data to an appropriate mathematical model.

For now, we will restrict attention to the following list of "core" functions:

Name	Formula	XFunctions Name	Important Points
Identity	x	none	-1	-1
			0	0
			1	1
Cube	x³	none	-1	-1
			0	0
			1	1
Cube Root		cubert	-1	-1
			0	0
			1	1
Greatest Integer	ëxû	floor	-1	-1
			0	0
			1	1
Absolute Value	\|x\|	abs	-1	1
			0	0
			1	1
Square	x²	none	-1	1
			0	0
			1	1
Square Root		sqrt	0	0
			1	1
			4	2

Throughout the text, we will add to this list, as we learn about new functions. Notice how all of these functions go through (0, 0), and (1, 1). Most of these functions also go through (-1, -1), and all of those are odd functions, except for the greatest integer function (Can you see why it is not symmetric with respect to the origin?). The remaining functions are even functions and so go through (-1, 1), except for the square root function. For the square root function, we choose (4, 2) as a third point, since it is the next easiest point to compute. Because these graphs go through so many of the same points, the most important thing to remember is the general shape of each graph, so we can "connect-the-dots" appropriately.

You should notice that we can deduce, from its graph, the formula of any function of the form, g(x) = ±a f(±(x - b)/c) + d, where f is any function in our list of "core" functions. We simply apply the same reasoning that led us to the point-slope formula. For example, consider the graph:

From the general shape of the graph, we should be able to determine which of the "core" functions f must be. This example looks like , but flipped, scaled, and shifted.

We can recognize that the point (-1, 1) on the graph of g corresponds to the "vertex" of f at (0, 0); this means that the graph has been shifted left 1 and up 1, so b = -1, and c = 1. Note: Even though the identity and greatest integer functions do not have a "vertex", per se, because they are symmetric with respect shifts along the 45° line, any reasonable choice will work; just as we saw that any point on a line will work, so any solid dot at the endpoint of a "step" of a greatest integer function will work, as well..

We can recognize whether or not a graph has been flipped. In fact, since all of these functions, except ëxû and are either even or odd, we can ignore horizontal flips and treat every flip as a vertical one; that is, we can ignore the ± sign "inside" f. In the case of ëxû, the flips are easy to spot by whether the "steps" go up or down, and whether the solid dots are on the left or right side of each "step". Likewise, any flips of can be seen in the way the graph curves and the direction it moves from its vertex. In this example, the graph is flipped vertically, so we know, so far, that g(x) = -a f((x - (-1))/c) + 1.

Finally, by identifying a point corresponding to (1, 1) on the graph of g, we can determine the scaling factors. In the previous section, we saw that the identify, square, and absolute value functions are "symmetric" with respect to scaling; that is, a horizontal and vertical scale are directly related. All of these "core" functions share this property (since they are all defined by powers; see the Note at the end of the previous section), with the exception of ëxû; this has the effect that we can choose any point to the right of the vertex to correspond with (1, 1). In our example, we can see that the graph goes through (2, -1). This means that we can view this graph as stretched vertically by a = 1 - (-1) = 2 (i.e., the vertical difference between this point and the vertex) and stretched horizontally by c = 2 - (-1) = 3 (i.e., the horizontal difference between this point and the vertex). That is, g(x) = -2f((x - (-1))/3) + 1. You can verify this with the following applet:

In general, we have an equation, which generalizes the point-slope formula:

If (x₁, y₁) corresponds to the "vertex" of the graph of a function, g, coming from one of the "core" functions, f, listed above, its equation may be written as:

g(x) = (y₂ - y₁)f((x - x₁)/(x₂ - x₁)) + y₁.

In this formula, for all "core" functions except ëxû, (x₂, y₂) is any other point on the graph; in the case of ëxû, (x₂, y₂) is a solid endpoint of the next "step" beyond the chosen "vertex".

Note: As before, the "flips" are included automatically, based on the relative positions of the two points!

Practice applying this formula to determine a formula from a graph by completing the following Exercises.