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

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

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

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 as

Linear graph

Linear.

This one?

This one?

Linear graph

Linear.

This one?

Nonlinear graph

Nonlinear.

This one?

Linear graph

Linear.

This one?

Nonlinear graph

Nonlinear.

This one?

Chicken

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:

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?

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

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

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.

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:

This graph is positive when x is less than 2 and negative when x is greater than 2.

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

Here's another:

This graph is positive when x is greater than -3 and negative when x is less than -3 and greater than 3.

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 + 3x - 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.

Lesson Summary

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 values increase. In decreasing functions, the 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.