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ELM: CSU Math Study Guide16 chapters | 140 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Functions do all kinds of fun things. In this lesson, learn how to identify traits of functions such as linear or nonlinear, increasing or decreasing and positive or negative.

Opposites - they're everywhere: yin and yang; cats and dogs; Republicans and Democrats; bacon and foods that just aren't bacon.

The idea of opposites also comes into play with functions. In this lesson, we're going to look at a few different kinds of opposites that matter for differentiating functions. Feel free to pet a cat or dog as you watch, or munch on bacon, just don't pet your cat with bacon. They don't like that.

First up, let's talk about linear or nonlinear functions.

A **linear function** is a function that represents a straight line. As you might expect, a **nonlinear function** is a function that represents a line that isn't straight. That's surprising, I know. But, that's really all it is. There are many ways of thinking about linear functions, but usually the simplest is to just remember that linear means line and nonlinear means, well, not a line.

If you're asked to identify a function as linear or nonlinear based on a graph, you're really just looking for a straight line.

This one?

Linear.

This one?

Nonlinear.

This one?

Linear.

This one?

Nonlinear.

This one?

Chicken.

If you just have the function and no graph, you can make a table. In fact, sometimes you'll be given a table of *x* and *y* values and asked if the function is linear or nonlinear. Here's one:

x | y |
---|---|

1 | 5 |

3 | 10 |

5 | 15 |

7 | 20 |

9 | 25 |

In a linear function, the *y* values will follow a constant rate of change as the *x* values. Above, notice that the *x* values are increasing by 2 each time. The *y* values are increasing by 5 each time. So, this is linear.

What about this one?

x | y |
---|---|

1 | 5 |

3 | 10 |

5 | 20 |

7 | 35 |

9 | 55 |

Here, the *x* values are going up by 2 again, but each time the *x* values go up by 2, the *y* values go up by different amounts. So, they're not constant, and this function is not linear.

Next, let's look at increasing or decreasing. Maybe your waistline is increasing as the bacon on your plate is decreasing.

To be **increasing**, a function's *y* value is increasing as its *x* value increases. In other words, if when *x*1 < *x*2, then *f(x*1) < *f(x*2), the function is increasing.

To be **decreasing**, the opposite is true - a function's *y* value is decreasing as its *x* value increases. In other words, if when *x*1 < *x*2, then *f(x*1) > *f(x*2), the function is decreasing.

An increasing function looks like this:

Here, when *x* is 0, *y* is -1. When *x* is 5, *y* is about 1. As *x* goes up, so does *y*. That's increasing.

Decreasing looks like this:

Here, the *y* values are getting smaller as the *x* values increase. When you have a graph like the one above, just think of increasing and decreasing as going up or down from left to right. If a line rises, it's increasing. If it falls, it's decreasing. You could also think of slope. A positive slope is increasing, while a negative slope is decreasing.

In a nonlinear function like this:

It's both increasing and decreasing. This one is increasing until *x* = 0 and decreasing when *x* is greater than 0.

If you were asked when this function is increasing, you'd say when *x* < 0.

Finally, let's look at positive or negative. As in, my dog has a positive outlook about everything, especially if it involves going for walks or smelling other dogs. Meanwhile, my cats have a negative outlook about things, especially things involving my dog.

A function is **positive** when the *y* values are greater than 0 and **negative** when the *y* values are less than zero.

Here's the graph of a function:

Where is it positive? When *x* < 2. And, it's negative when *x* > 2.

Here's another:

This one is positive when *x* > -3 or *x* < 3. It's negative when *x* is < -3 and > 3.

Of course, you can do this without the graph. Let's consider *f(x)* = x^2 + 3*x* - 2. Is it positive or negative when *x* = -1? Just plug in -1 for *x*. So, *f(x)* = (-1)^2 + 3(-1) - 2. That's 1 - 3 - 2, or -4. -4 is negative, so *f(x)* is negative when *x* = -1.

What about when *x* = 3? So, *f(x)* = (3)^2 + 3(3) - 2, which is 9 + 9 - 2, or 16. 16 is positive, so *f(x)* is positive when *x* = 3.

In summary, we learned about opposites.

There are **linear** and **nonlinear functions**. Linear functions represent straight lines, while nonlinear functions are lines that aren't straight.

There are **increasing** and **decreasing functions**. In increasing functions, the *y* values increase as the *x* values increase. In decreasing functions, the *y* values decrease as the *x* values increase.

Finally, there's **positive** and **negative**. This just means is *y* positive or negative for a given *x* value?

As for cats and dogs, well, I guess we didn't learn much about them, but that's OK.

Following this lesson, you should be able to:

- Differentiate between linear and nonlinear, increasing and decreasing, and positive and negative functions
- Explain how to identify these different functions with and without a graph

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ELM: CSU Math Study Guide16 chapters | 140 lessons

- Graph Functions by Plotting Points 8:04
- Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative 5:49
- How to Find and Apply The Slope of a Line 9:27
- How to Find and Apply the Intercepts of a Line 4:22
- Graphing Undefined Slope, Zero Slope and More 4:23
- Equation of a Line Using Point-Slope Formula 9:27
- How to Use The Distance Formula 5:27
- How to Use The Midpoint Formula 3:33
- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- How to Graph Cubics, Quartics, Quintics and Beyond 11:14
- Go to ELM Test - Geometry: Graphing Functions

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