Linear Functions

Linear Functions are defined to be polynomials of degree no greater than \(1\). More often than not, they are written in the form \(f(x)=mx+b\) where \(m\) and \(b\) are real numbers (note how this is the same as slope-intercept form \(y=mx+b\)). \(m\) is the slope of the line, and \(b\) is the \(y\)-intercept, or where the line crosses the \(y\)-axis when graphed.

Linear Equations are equations involving polynomials of degree no greater than \(1\).

Linear functions have a wide variety of uses. More often than not, they’re used for establishing trends on complicated, zig-zaggy, graphs and charts to show generally where a data-set is headed next, or to describe/model growth or decay over time in real-world applications.

The most useful aspect of a linear function is its rate of change or slope, which can be used to describe, on average, how much a quantity is rising or falling over a certain time frame (or with respect to another quantity). This is useful when you are looking for a “gist” for where a certain value is headed over time (e.g. think “number of sales” over “time”), when the graph given to you with specific data is quite ugly or chaotic.

The blue graph above represents the price of stock in a particular company. The green line has been placed such that it touches the lowest points on this graph. Thus, if the pattern continues for this stock, the slope of the green line can be used to predict a theoretical “lowest possible price” of the stock at some future time.

All in all, lines are the easiest sort of function type for humans to think about. We want everything to be in nice, straight, even lines so that we can draw conclusions quickly about whatever data we are looking at. So, in all the lessons that follow, we will discuss everything you need to know about linear functions and linear equations!