Linear vs. Nonlinear

Now, how do we know if a function is linear or non-linear? A linear function is defined as a first-degree polynomial. This simply means that the exponent of the x is 1. A linear function can also be written in the slope-intercept form we've discussed, y = mx + b where m represents the slope and b represents the y-intercept.

Here are some examples of equations that are linear functions:

y = 2x + 1
y = -1/2x - 4
y = x

Here is an example of a linear function graph (notice it is a straight line):

image of negative straight line on graph

Here is an example of a linear function written in a table:

x	0	2	4	6
y	1	2	3	4

Another way to determine whether a function is linear is to look at the equation. A linear equation that can be written in the standard form.

Standard Form of a Linear Equation: Ax + By = C, where A, B, and C are real numbers and A and B are not both 0.

When a linear equation is written in standard form:

X and y both have exponents of 1

X and y are not in denominators, exponents, or radical signs

X and y are not multiplied together

table contrasting linear and nonlinear functions

Graphing linear equations is simple because since all your points are on one straight line, you only need two points to graph the linear equation.

Using the standard form, you could easily substitute 0 for x and solve for y, giving you one ordered pair. Then, go back and substitute 0 for y and solve for x, giving you the second ordered pair. Finally, use a ruler to draw a straight line that passes through these two points.

Using the slope-intercept form of the linear equation (y = mx + b)to graph the line, simply plot your first point at the y intercept (b) where x is 0. Next, look at your m value and use the rise over run strategy to find the next point. If m is 2, you would start at point b, then move up two spaces and to the right 1 space to get to the next point.

There is one more form that could be used to graph a linear equation. If you know the slope and any point on a graph, you can graph the line using the Point-Slope Form.

LaTeX: y-y_1=m(x-x_1)

This formula represents slope(m) and one point on the line (x₁,y₁) .

Linear and Nonlinear Practice

If you would like to practice more problems and check your work, click here. Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand.

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