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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you'll be able to tell the differences between linear and nonlinear functions. You'll also be able to identify both types just by looking at graphs.

What is the one thing that you end up working with the most in math? Functions! In this lesson, you will learn the two types of functions that are out there in the world of math, including the differences between the two and what kinds of equations you can expect to see for both.

The two types are linear functions and nonlinear functions. **Linear functions** are the functions that graph out to a straight line. **Nonlinear functions** are the functions that don't graph out to a straight line. Why should you learn this? Because you'll be able to solve problems much more easily if you know what kind of function you're dealing with. This is so because you will have different formulas for each type of function. Some formulas are for working with linear functions only and some other formulas are for working with nonlinear functions only. So, being able to spot the type of function you are working with right away will help you solve your problems faster.

Let's get started.

As the name implies, linear functions are functions of lines; straight lines, to be exact. This graph here shows what a linear function looks like when graphed. All linear functions will graph out to a straight line. The location of the line may be different and the slant may be different, but the line will always be straight. Just tell yourself that if you can draw the function's line by making just one line with a ruler, then it is linear.

Nonlinear functions, on the other hand, graph out to anything but a straight line. Nonlinear functions can graph out as curves, two straight lines that meet somewhere, or circles or ellipses. Anything but a straight line is possible with nonlinear functions. All of these graphs here are examples of nonlinear functions.

You can't draw any of these with just one line from a ruler. You either have to move your ruler or you have to use another tool to draw the curves.

The equations for both types are also different. What you can expect to see in one, you can't expect to see in the other. Both linear and nonlinear functions use variables, typically *x* and *y*, but how the equations are written are very different.

Linear functions can come in any one of three forms. The first form is called the **slope-intercept form**. This form is written *y* = *mx* + *b*, where *m* is the slope, or slant of the line, and the *b* is the *y*-intercept, or where the line crosses the *y*-axis.

The second form is called the **point-slope form**. This form is *y* - *y* sub 1 = *m*(*x* - *x* sub 1), where *y* sub 1 and *x* sub 1 are the *y* and *x* coordinates of a point on the line and the *m* is the slope.

The third form is called the **general form**. It is written as *Ax* + *By* + *C* = 0, where *A*, *B*, and *C* are numbers, and with *A* and *B* never both being 0. If *A* is 0, then *B* cannot be 0 and vice versa.

Looking at these three forms, you will notice that the *x* and *y* variables are always written as *x* and *y*. Of course, you can have variables other than *x* and *y*. They can actually be any two letters. Regardless of which letters they are, they are never written with exponents or inside a root of any kind, nor are they written in the denominator. Linear functions will never have exponents or roots associated with the variables.

*y = 2x - 3y - 3 = 4(x - 10)4y + 5x + 9 = 0y / 3 = 6x = 3y = x*

Nonlinear functions, however, will. You will see nonlinear functions where the *y* or the *x* is squared. You will see other exponents as well. You can also see square roots or third roots as well. You may also see the variable in the denominator.

*y = x^2sqrt(y) = 4xx^2 + y^2 = 16y + x^3 = 6y = 1 / x*

Let's review what you've learned now.

**Linear functions** are the functions that graph out to a straight line. **Nonlinear functions** are the functions that don't graph out to a straight line. The graph of a linear function is always a straight line. The graph of a nonlinear function is anything but a straight line.

Linear functions come in three forms. The first form is called the **slope-intercept form**. This form is written *y* = *mx* + *b*, where *m* is the slope, or slant of the line, and the *b* is the *y*-intercept, or where the line crosses the *y*-axis. The second form is called the **point-slope form**. This form is *y* - *y* sub 1 = *m*(*x* - *x* sub 1), where *y* sub 1 and *x* sub 1 are the *y* and *x* coordinates of a point on the line and *m* is the slope. The third form is called the **general form**. It is written as *Ax* + *By* + *C* = 0, where *A*, *B*, and *C* are numbers and with *A* and *B* never both being zero.

Linear functions will never have exponents or roots associated with the variable, nor will the variable be in the denominator. These are all the possibilities with nonlinear functions.

Once you are finished, you should be able to:

- Name and describe the two types of functions in math: linear and nonlinear
- Recognize the graph of a linear and a nonlinear function
- Determine if the graph of an equation will be linear or nonlinear

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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

- What is a Function: Basics and Key Terms 7:57
- Transformations: How to Shift Graphs on a Plane 7:12
- How to Add, Subtract, Multiply and Divide Functions 6:43
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- How to Compose Functions 6:52
- Inverse Functions 6:05
- Applying Function Operations Practice Problems 5:17
- Linear and Nonlinear Functions 5:56
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