Linear vs Non-Linear Relations

Differences between Linear and Quadratic Equations

  • Qudratic functions are expressed as \(y = x^2\) while linear expressions are expressed as \(y = x\)
  • Quadratic functions have \(2\) as the degree of their highest term whereas linear functions have \(1\) as their highest degree
  • All quadratic functions increase and decrease regradless of the slope whereas linear equations either always increase (if the slope is positive) or decrease (if the slope is negative)
  • The slope of a quadratic function is constantly changing. The slope of a linear function is always constant
  • Linear functions have each input produce a unique output. Conversely, quadratic functions (with the exception of the vertex) have pairs of unique independent variables produce the same output (ie \(x\) values of \(2\) and \(-2\) would produce the same result)

Scatter plots can also be used to identify the different relations between \(x\) and \(y\):

  • The relationship is considered linear when the points on a scatter plot follow a somewhat straight line pattern
  • The relationship is considered non-linear if the points on a scatter plot follow a pattern but not a straight line



Identifying Linear and Non-Linear Equations

As stated above, a function can be determined by the degree of their highest term - also known as the Leading Term (ie a Linear Function's highest term is \(1\) while a Quadratic Function's highest term is \(2\)). You will learn about other non-linear relations if you continue studying math!

Try and fit the following functions into their correct description:

\(x^2 + 2\)

\(2x + 6\)


\(x^2 + 5x + 6\)



\((6x + 1)(x - 4)\)

\(8(x-5)^2 + 3\)