Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative

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  • 0:06 Opposites
  • 0:34 Linear or Nonlinear
  • 2:01 Increasing or Decreasing
  • 3:36 Positive or Negative
  • 5:09 Lesson Summary
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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.

Linear or Nonlinear

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 graph
linear graph


This one?

Nonlinear graph
nonlinear graph


This one?

Linear graph
linear graph


This one?

Nonlinear graph
nonlinear graph


This one?

picture of 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.

Increasing or Decreasing

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 x1 < x2, then f(x1) < f(x2), 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 x1 < x2, then f(x1) > f(x2), the function is decreasing.

An increasing function looks like this:

Graph of increasing function
graph of increasing function

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:

Graph of decreasing function
graph of decreasing function

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:

This nonlinear function is both increasing and decreasing.
graph of nonlinear function

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.

Positive or Negative

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:

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