The easiest way to determine a linear function is by observing the way that it’s been graphed. If it’s a straight line, then it is a linear function. There’s more to it than that, of course. In this guide, we’ll go over some linear function examples to help you better understand the logic and application of linear functions.

In mathematics, linear functions are used in algebra and calculus. Check out this entry-level Algebra I course, or this course on essential concepts in calculus to get started.

## What is a Linear Function?

When something is linear, it means it progresses from one stage to the next in a straight, sequential fashion. No winding around, no fancy zig-zags, and no confusing paths. Just a simple, straight line.

In mathematics, when see a straight line plotted and drawn on a graph, you’re seeing a linear function. However, there are ways to recognize linear functions without looking at its graph. If a function is linear, then the change in the *y* coordinate must always be consistent, in relation to the change in *x*. This is done by plugging coordinates into the following equation:

### f(x) = ax + b

Here, *a* is the slope of the line, and *b* is the *y*-intercept of the line.

This is most useful for calculating the rate of change when specific coordinates are not given. Before we get into that, let’s see how this function even works.

## Recognizing a Linear Function

Before we get started, consider reviewing this quick refresher on basic algebra concepts.

Let’s say we’re looking at a set of coordinates, and we want to determine if the graph will be linear. These coordinates indicate the number of gallons of liquid in a large vat, in the *y* coordinate, and the number of minutes in increments of ten that have passed since the vat began to drain, in the *c* coordinate.

x = minutes passed y = gallons of liquid in vat x | y --------------- 0 | 500 10 | 450 20 | 400 30 | 350 40 | 300 50 | 250 60 | 200

Now, just from looking at this graph, you might be able to tell that it’s a linear one. This is because I’ve chosen simple numbers, but this might not always be the case. Just to be safe, we’ll plug the coordinates into the function and double check to make sure we’re right.

In order to do this, though, we need to calculate slope.

To calculate a line’s slope, you divide the change in *x* by the change in *y*, which requires at least two of each coordinate.

*x _{1}* = 0 and

*x*= 10

_{2}*y*= 500 and

_{1}*y*= 450

_{2}To get the slope, we’ll calculate *y _{1}* –

*y*/

_{2}*x*–

_{1}*x*

_{2}That’s (500 – 450) / (0 – 10) = 50/-10 = -5

So the slope of this function is -5.

To find the *y*-intercept, we just need to plug in one instance of *y* and *x*.

If *y* = *ax* + *b*, we can say:

500 = -5(0) + b

500 = 0 + b

b = 500

The *y*-intercept of our function is 500. Our function is linear, with no negative numbers in the *x* coordinate, so this is actually a given, but it is good and necessary to know how to calculate this.

So, let’s calculate that linear function already!

gallons of liquid in vat = number of minutes passed × slope + y-intercept

Remember, we’re working with:

x = minutes passed y = gallons of liquid in vat x | y --------------- 0 | 500 10 | 450 20 | 400 30 | 350 40 | 300 50 | 250 60 | 200

500 = 0(-5) + 500

500 = 0 + 500

500 = 500

450 = 10(-5) + 500

450 = -50 + 500

450 = 500 – 50

450 = 450

Success! Our function is, indeed, linear.

Here’s what it looks like graphed!

To graph your own custom coordinates, check out this website. For a step-by-step guide on how graphing in algebra works, check out this entry-level graphic course.

## Calculating Coordinates with a Linear Function

If this is all familiar to you, check out this intermediate algebra course for more information about linear functions.

So, we know the coordinates that we have will graph a linear line on a chart, but we don’t just graph charts to make sure the lines are straight or not. We want to recognize patterns of data. Using our slope and *y*-intercept, we should be able to plug in numbers for either *x* or y and determine things like how many minutes need to pass for our vat to be at 0 gallons. Let’s test now.

0 = -5(x) + 500

Here, we’ve plugged in 0 in place of *y*, and we’ll be solving for *x*.

-500 = -5x

5x = 500

x = 100

This means that when *y* is 0, *x* will be 100. It will take 100 minutes for our vat of liquid to drain completely.

## Linear Function Examples

Need more coordinates to work with? Check out the ones below, and see if you can determine their linearity!

x | y --------------- 1 | 5 2 | 10 3 | 15 4 | 20 5 | 25 6 | 30 7 | 35 x | y --------------- 3 | 125 6 | 127 9 | 129 12 | 131 15 | 133 18 | 135 21 | 137

Check out this course for more on advanced functions. If you’re unfamiliar with the concepts explained here, you might want to take a step back and try out this foundational algebra course for beginners.