# Recognizing Linear Functions

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A function describes a relationship in which each $x$-value corresponds with exactly one $y$-value. There are different ways to group functions based on certain characteristics or qualities they have. Here, how a function’s rate of change can make it unique, will be explored.

## Rate of Change

The rate of change, ROC, is a ratio that describes the **average change** between two related points. It can represent the average speed of a car over a certain period of time, or the average growth rate for bacteria in an experiment. It's determined by dividing the change in the vertical direction ($y$) by the change in the horizontal direction ($x$).

$\text{rate of change}=\dfrac{\text{change in } y}{\text{change in }x}$

The Greek letter $\Delta$ (Delta) is commonly used to describe a difference, which leads to an alternative notation of the formula.

$\text{rate of change} = \dfrac{\Delta y}{\Delta x}$

## Constant Rate of Change

If the rate of change of a function is constant, meaning that for all points on the function, the ROC is always the same, the function is said to be linear. By determining whether or not the rate of change is constant, it's possible to separate linear functions from non-linear functions.

## Linear Function

If a function has a constant rate of change, it is linear. Graphically, a linear function is a straight line.

Using that line, it's possible to determine the rate of change by finding the horizontal change $(\Delta x)$ and the vertical change $(\Delta y)$ between any two given points on the line. Any function whose graph is not a straight line cannot be linear.## Exercises

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