To determine whether a function is linear or not, we need to verify that the equation is a polynomial of the first degree. This means that the function must have the form when reorganized, and the independent variable *x* must have an exponent of 1. We will learn more about this with some exercises.

## What are linear functions?

A linear function is an algebraic equation, in which each term is either a constant or the product of a constant and a variable (raised to the first power). For example, the equation is a linear function since both variables *x* and *y* meet the criteria, and both constants *a* and *b* do as well.

The exponent of *x* is 1, that is, it is raised to the first power and the equation follows the definition of a function: for each input (*x*) there is only one output (*y*). The graph of a linear function is a straight line.

### EXAMPLES

The following are linear functions:

## Graphs of linear functions

The origin of the name “linear” comes from the fact that the set of solutions of this type of function forms a straight line in the Cartesian plane.

On the graph of a linear function , m determines the slope of that line, that is, the steepness, and *b* determines the *y*-intercept, that is, the point where the line crosses the *y*-axis.

If you need help with this topic, check out our guide on how to graph linear functions.

### EXAMPLES

The following is the graph of :

The following is the graph of :

The following is the graph of :

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## How to know if a function is linear?

A linear function creates a straight line when graphed on the Cartesian plane. Therefore, if it is possible to graph the function, you can easily determine if a function is linear by checking that its graph produces a straight line.

To know if a function is linear without having to graph it, we need to check if the function has the characteristics of a linear function. Linear functions are polynomials of the first degree.

Verify that the dependent variable or *y* is by itself on one side of the equation. If it is not, rearrange the equation to isolate the dependent variable. For example, if we have the equation , we move the to the right to get and then divide the entire equation by 4 to get .

Now, we have to verify that the equation formed is indeed a linear function. For an equation to be a linear function, the equation must be a polynomial of the first degree. This means that the power of the independent variable or* x* must be 1. In our example, we see that *x* has a power of 1, thus, is a linear function.

## Examples to determine if a function is linear

### EXAMPLE 1

Determine whether the equation is a linear function.

**Solution:** We can easily see that the variable *x* has a power of 1, therefore, the equation represents a linear function. We can verify this by plotting its graph:

We see that we obtained a straight line, so the equation is a linear function.

### EXAMPLE 2

Determine whether the equation is a linear function.

**Solution:** We can isolate the dependent variable to facilitate the visualization of the equation. Then, we have:

We see that we have the variable *x* with a power of 1, which means that it is a linear function. Looking at its graph we have:

The graph is a straight line, so the equation is a linear function.

### EXAMPLE 3

Determine whether the equation is a linear function.

**Solution:** In this case, we see that the variable *x* has a power of 2. This means that the function is not linear. This is a quadratic function. Let’s look at its graph to verify this:

We see that we obtained a parabola, thus, the equation is not a linear function.

### EXAMPLE 4

Determine whether the equation is a linear function.

**Solution:** We have to simplify and rearrange the equation for easier visualization:

Clearly, the equation is not a linear function since the variable *x* has an exponent of 2. Let’s look at the graph of the function:

We see that we obtained a parabola, so the equation is not a linear function.

## See also

Interested in learning more about functions? Take a look at these pages:

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