Nonlinear Functions and Their Graphs

What are the features of a linear graph? Not much, it has a slope, x and y intercepts (and only one of each). After that, a line has little to distinguish it.

Now think about nonlinear functions. Potentially they have many features.

First we notice that they can have ups and downs. We call the highest elevations the maximum values and the lowest elevations the minimum values. When we speak generally about the size (elevation) of the maximums and minimums, we talk about the extremes (extreme values) of the graph.

Here's where we have some problems in vocabulary. When we talk about the extreme locations, we are focused on the input (x values). When we care how big or small the extreme is we talk about the extreme values (y values).

The graph to the left shows three different cubic functions. Click on the graph to get a more readable image. They have very different maximum and minimum extreme values. However, those values occur at the same location in their mutual domain. The "How big?" part is only important after we answer the "Where is the maximum value?" question.

If all values are equal we do not discuss extreme value. Here's a simple one-dimensional example:

Look at a horizontal number line. In the example below, the only location information we have is the x coordinate. There is no extreme value because all of them have equal elevation.

While it may be fair talk about the largest value or smallest value in the left to right locations, we should reserve the idea of extreme values to situations where we have vertical changes.

Now what happens when we have a two-dimensional situation? Consider the next graph. It has all the features that are fit to talk about in the world of extremes.

Look at each of the red dots. Each represents a high point on the graph. They are all relative maximums. The term relative means that each is the "highest" in its area of the graph. Of these three points, the one at (0, 5) is highest of all. It is the absolute maximum. It can correctly claim "the" highest. The graph has an absolute maximum value of 5. The absolute maximum value is located at x = 0. (The term local and global are also used for extremes.)

Now look at the blue dots. Each represents a low point on the graph. They are all relative minimums. Again, the term relative means that each is the "lowest" in its area. Of these three points, the one at (5, -3) is lowest of all. It is the absolute minimum. It is "the" lowest. The graph has an absolute minimum value of -3. The absolute minimum value is located at x = 5.

Let's go back to the concept of when a function is increasing or decreasing. In the context of extremes, a function increases moving left to right from a relative minimum to a relative maximum. A function decreases moving left to right from a relative maximum to a relative minimum.

Even Functions, Odd Functions and Those that are Neither

The terms even and odd tell us about the symmetries of a graph.

Graphs that are even have symmetry with respect to the y-axis.

Graphically we say the graph appears unchanged when reflected across the y-axis.

Algebraically that means that . The graph to the left demonstrates even symmetry.

Graphs that have odd symmetry have symmetry with respect to the origin.

Graphically we say that when the graph is rotated 180 degrees about the origin, it appears unchanged. This a is a "pin wheel effect"

Algebraically that means that . The graph to the right has odd symmetry.

If a graph doesn't have even or odd symmetry, it has neither symmetry. So it fails both of the tests above. The graph to the left has no symmetry.

Question: Is there any polynomial function with a graph that has both even and odd symmetry simultaneously?

Question: Is there any function with symmetry with respect to the x-axis? But any other graph that has this symmetry fails the VLT since some x value must create two y values.

End Behaviors (Left and Right Hand Behaviors)

We could subtitle this "A Tale of Two Tails." The idea of end behaviors is to decide how a function acts when we put in values for x really distant from the origin on the x-axis.

Here we also need to be careful about vocabulary. What do "really large" and "really small" mean?

Really large values for x are easy. Ten million, ten billion, a billion billions are all large to us.
However going the other way from the origin, negative ten billion looks large, but is very small. Relative to that number, zero is large!
Sometime we say "large in absolute value" to make sure others know that we're talking about distant from the origin.

We have symbology to represent the ideas. (Of course, we do!)

To write "far from the origin on the positive x-axis," write . We say "as x gets very large" or "as x goes to infinity", or "x increases without bound."
Then, to write "far from the origin on the negative side," write . We say "as x gets very small" or "as x goes to negative infinity", or "x decreases without bound."

Finally we can talk about how functions behave under these two conditions: or .

End behavior is describing a function's past and predicting its future based on characteristics of its rule of calculation. We look from the present (the origin) to the left and right to decide behaviors. Terms like rising and falling are often applied.

Please don't confuse rising and falling with increasing and decreasing.

Increase and decrease are always read from left to right on the number line.
Rise and fall are descriptions applied from the origin to the infinities, and .

Polynomials are the easiest of all to talk about. We only need to know the highest degree and sign of the leading coefficient. We often say the leading term is the dominant term. Eventually it has almost complete control of the calculation.

Example: Describe the end behavior for y = x².

We know what the graph looks like. It's the big smile. We also know the function has a single term, x², and only produces nonnegative values for y. The term has a positive sign. So the polynomial function has a positive quadratic dominant term.

Looking left, as we see that . We say "as x decreases without bound, y increases without bound."

Looking right, as we see that again . We say "as x increases without bound, y increases without bound."

Notice that the polynomial function has an even highest degree. We expect the end behaviors to be the same. Both increase without bound.

Example: Now think about y = x² + x. It has the same dominant term, so it has exactly the same end behaviors. The linear term just cannot compete with the squaring action when x is large in absolute value.

Notice that both of the quadratic functions above have an even highest degree. The end behaviors are the same.

Example: Describe the end behavior for y = x³.

We know what the graph looks like. In the old vocabulary the function increase across its domain. In the new terminology, it falls to the left and rise to the right from the origin. We also know the function has a single term, x³, and produces values for y with both negative and positive signs. The term has a positive sign. So the polynomial function has a positive cubic dominant term.

Looking left, as we see that . We say "as x increases without bound, y decreases without bound."

Looking right, as we see that . We say "as x increases without bound, y increases without bound."

Example: Now think about y = x³ + x². It has the same dominant term, so it has exactly the same end behaviors. The quadratic term just cannot compete with the cubing action when x is very large in absolute value.

These cubic polynomial functions have an odd degree in the dominant term. The end behaviors are opposites.

When the leading coefficient is negative the end behaviors reverse. Graph the following to observe their end behaviors.

	**y = - x² + x**	*y = - x³ + x²*	**y = - x² - x**	*y = - x³ - x²*
as
as

End Behaviors of Non-linear, Non-polynomial Functions

The ideas is the same but the result can be quite different. For these functions, we are looking for horizontal asymptotes. This term just tells us that instead of going off to some infinity, we approach a constant value.

Let's look at a few examples.

Example 1:

The graph is to the right:

Looking left, as we see that . We say as "x increases without bound, y approaches a horizontal asymptote at y = 0. "
Looking right, as we see that also. So, as "x decreases without bound, y approaches a horizontal asymptote at y = 0."

Notice that the function has a value of zero only at the origin. It is never zero again.

Look at the numerator and the denominator. The dominant term is in the denominator. It takes control and gets bigger so much faster than the numerator that it takes the whole relationship to zero.

We'll do the rest by graph only.

Example 2:

Click on the graph for a better image.

Looking left, as we see that . We say "as x increases without bound, y approaches a horizontal asymptote at y = 0.
Looking right, as we see that also. So we say, "as x decreases without bound, y approaches a horizontal asymptote at y = 0."

Warning: Notice that the "y = 0" in the context of these statements is talking about a horizontal line. So this is the description of a specific constant function. While this constant function provides a visual representation of the horizontal asymptote, it is not a part of the graph.

This is a subtle interpretation, but the graph is created by . It can never be zero.

Example 3:

Click on the graph for a better image.

Looking left, as we see that . We say "as x increases without bound, y approaches a horizontal asymptote at y = 0.
Looking right, as we see that also. So, "as x decreases without bound, y approaches a horizontal asymptote at y = 0."

This is really subtle interpretation. This called a damped sine curve. While it crosses zero infinitely many times, its end behaviors are to wiggle across y = 0 forever getting closer and closer to it. If you want to explore this more, enter at y = sin(x)/x into your calculator. Then use table values like x = 10^6 and -10^6. Sine functions are not a part of this course, they just provide nice examples.

We will study more about end behaviors later.