Linear functions

A linear function is any function that is given by y = ax + b, where a and b are real numbers. A special case of a linear function occurs when a = 0, since the previous notation can be abbreviated as follows: y = b, which is a constant function (some sources do not count constant functions as linear functions).

Types of linear functions

Linear functions have rather nice graphs because they are always described by a line, for example, the constant function already mentioned has a graph in the form of a line parallel to the x-axis, intersecting the y-axis at the point b. If b = 0, the straight line always passes through the origin [0, 0]. This function is also referred to as a linear proportionality.

Other properties of a linear function arise from what a. For if a > 0, it is a graph of an increasing function, but if a < 0, the graph is a decreasing function. The graph of the function y = ax will be axisymmetric along the axis y with the function y = −ax.

Properties of linear functions

Thedefining domain of linear functions is the set of real numbers, as with the domain of values. The function is decreasing or increasing depending on a constant a. It is a simple function, since we cannot find a horizontal line that intersects the graph of a linear function at more than one point (not true for a constant function). Furthermore, it is not periodic, it is continuous over its entire defining domain, it has no maximum or minimum. A linear function is neither even nor odd, only if b = 0, it is an odd function.

Example

Draw the graph of the function y = −3x +1.

The graph of this function is easy to draw. We know that the graph of any linear function is a line (theoretically, it can also be a line segment if you are to plot the graph of a linear function only on a certain interval). That will be plenty for us. To plot a straight line we only need to know the coordinates of just two points. Let's calculate them. Let's first get the simplest number, zero, after x. We get f(0) = 1. The first point that the line will pass through will be [0, 1]. For the second point, we can put, for example, one. We get this: f(1) = −3 + 1 = −2. The second point will have the coordinates [1, −2]. We now have the two points needed to plot the graph of this function: