How to Know If a Function is Linear?

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 f(x)=mx+b when reorganized, and the independent variable x must have an exponent of 1. We will learn more about this with some exercises.

ALGEBRA
how to know if a function is linear

Relevant for

Learning to determine if a function is linear or not.

See method

ALGEBRA
how to know if a function is linear

Relevant for

Learning to determine if a function is linear or not.

See method

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 y=ax+b 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:

  • f(x)=4x+3
  • y=2x-5
  • x+2y=4
  • f(x)=\frac{1}{3}x+6

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 f(x)=mx+b, 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 f(x)=2x-2:

graph of a linear function 1

The following is the graph of y=-x+3:

graph of a linear function 2

The following is the graph of f(x)=\frac{1}{3}x+1:

graph of a linear function 3

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 3x+4y=4, we move the 3x to the right to get 4y=-3x+4 and then divide the entire equation by 4 to get y=-\frac{3}{4}x+1.

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, y = -\frac{3}{4} x + 1 is a linear function.


Examples to determine if a function is linear

EXAMPLE 1

Determine whether the equation y=3x+2 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:

graph of a linear function 4

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

EXAMPLE 2

Determine whether the equation 2x-2y=6 is a linear function.

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

2x-2y=6

-2y=-2x+6

y=x-3

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:

graph of a linear function 5

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

EXAMPLE 3

Determine whether the equation y=2{{x}^2}-3 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:

graph of a parabola 1

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

EXAMPLE 4

Determine whether the equation 3x(2x+1)+2y=5 is a linear function.

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

3x(2x+1)+2y=5

6{{x}^2}+3x+2y=5

2y=-6{{x}^2}-3x+5

y=-3{{x}^2}-\frac{3}{2}x+\frac{5}{2}

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:

graph of a parabola 2

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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