Nonlinear Functions in Algebra

Functions are a way to understand how one quantity grows or decreases compared to another quantity.  Nonlinear functions, which we will discuss here, are functions that have different growth rates.

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Introduction

We have already talked about linear functions, where growth rates are the same. So, when growth rates change, we call this nonlinear growth. Nonlinear growth happens in many situations. Two ways in which we can see nonlinear growth are in quadratic and exponential functions. We have already talked about quadratic functions, so this should not be surprising.

Polynomials

Polynomial functions are useful, especially in graph form. In graph form, it is easy to see how a function be haves and what it means. Polynomials deal with real numbers and its graph will be continuous. We can describe them by their characteristics:

• Formulas
• Degrees
• Coefficients

When looking at the formula, we can see a lot about the function. The degree of a polynomial is directly related to the variable exponent, like x^2 or x^3. An x^2 function is a degree 2, for example. Coefficients change function behaviors in strange and interesting ways. I will get to that later.

Functions that are degree two or higher are nonlinear. Hopefully, you can start seeing the relationships between these different functions. I should also point out that any function with a radical, ratio, or absolute value is not a polynomial.

Is this a polynomial:

$f(x) = 2x^3 -x +5$

Yes this is a polynomial because it does not contain radicals or complex numbers.

Is this one a polynomial:

$f(x) = \sqrt{x}$

This one is not a polynomial because it contains a root.

Increasing and Decreasing Functions

When looking at a graph of a function, it can stay constant, increase, or decrease. You already know what a constant function is. So, a function that is increasing is going uphill from left to right. Conversely, a function that goes downhill, from left to right, is a decreasing function.

We can also look at a function over a certain interval. It can be the entire interval of a function or just a small piece of it. For example, if we had a report on the revenue of a company over several years, we could evaluate the profits for certain years. This would look at a small interval of a function to see if it was increasing or decreasing.

Is this function increasing or decreasing?

$f(x) = -3$

This function is neither increasing or decreasing.

Is this function increasing or decreasing?

$f(x) = 2x - 1$

This function is increasing and it never decreases.

Symmetry in Functions

Symmetry in functions is an important topic later on. For now, it can tell you whether or not a function is even. A function is symmetrical if it is the same on either side of an axis. We can see symmetry in nature and we often think of things that are symmetrical as beautiful. The same is true in math. We look for it because it tells us attributes about a function.

A function is even if it is symmetrical around the y-axis. The function is odd if it is symmetrical around the x-axis. We can also tell this by looking at the equations themselves. If a function only has even exponents, it is an even function. This means that a function that contains only odd exponents is an odd function.

Another important fact is that many functions and their graphs have no symmetry at all. This is completely normal, so you should expect it.

Nonlinear functions can increase or decrease when represented graphically. This can change depending on the interval. For example, a function can be both increasing or decreasing, depending on the interval. When these functions are degree two or higher, they are then nonlinear.

Is this function even, odd, or neither?

$f(x) = 5x$

This is an odd function.

Is this function even, odd, or neither?

$f(x) = x + 3$

This function is neither even or odd.

Is this function even, odd, or neither?

$f(x) = x^2 - 10$

This would be an even function.

Polynomial Functions

Historically, these have greatly interested mathematicians. The reason why is that they are very useful in modeling data. We use them all the time for this. Computer science, mathematics, data science, and several other disciplines use them regularly.

Polynomials in graphs are continuous. Their lines have no breaks. Since they include only real numbers, radicals and complex numbers disqualify them. Their coefficients must also be real numbers.

We can break down polynomials. Different forms of a polynomial mean different things.

• function —— f(x) = x^2 + 2x + 2
• polynomial——x^2 + 2x + 2
• equation——x^2 + 2x + 2 =0

It will be important to know these differences.

Another important definition, a turning point, is when a graph changes from decreasing to increasing. A turning point is defined as the point in which this happens.

As you know, polynomial functions can have many different levels of exponents. The exponents mean a lot, graphically. A function with an x^3 term will have three intercepts and 2 turning points. An x^7 function will have 7 intercepts and 6 turning points. So, you can look at a function and tell a lot about it. A parabola crosses the x-axis twice, so it has an x^2 term, which we already know.

What is the degree and leading coefficient of:

$f(x) = -2x + 3$

The degree is 1 and the leading coefficient is -2.

What is the degree and leading coefficient of:

$f(x) = x^2 + 4x$

The degree is 2 and the leading coefficient is 1.

What is the degree and leading coefficient of:

$f(x) = -2x^3$

The degree is 3 and the leading coefficient is -2.

Piecewise Polynomials

You can have piecewise polynomial functions. In order for this to happen, each separate part of the function has to be a polynomial. You treat each part as a separate part unless asked to do something unique.

Conclusion

Polynomial functions that are at least second degree have more complicated graphs than a constant function. You can tell the x-intercepts and turning points from the shape of a graph. This makes the study of graphs beneficial. As previously stated, nonlinear and polynomials have a continuous graph. They also are made of real numbers and have real number coefficients.

Thank you for reading this, I really appreciate it.

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