# Linear Functions and Parameters

## Overview

Let's look more at the **symbols** in linear functions. of the form $f(x) = mx + b$. The **m** and **b** are **parameters**, which are substituted with numbers to get a specific example of a function.

:bulb: In general, anything in a function besides the input variables and output variables are parameters. 

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## Is the Relationship Linear?

Before we start trying to find the equation of a line, it's good to determine if the relationship between the variables is actually linear!

:bulb: In a linear equation, the *rate of change* is always the same.

**Formula:** We calculate the rate of change on an interval from $x = a$ to $x = b$ as $\displaystyle\frac{f(b) - f(a)}{b-a}$.

You may have seen this before $\displaystyle\frac{\Delta y}{\Delta x}$ or anything similar.

Also, we can reason through this process by asking "If we change changing $x$ by fifty times as much, will it change $y$ by fifty times as much?" (This [heuristic](https://en.wikipedia.org/wiki/Heuristic) is suggested by the philosopher [Nassim Nicholas Taleb](https://www.fooledbyrandomness.com/))

* Example: We sell widgets for $2 a widget. If we sell one widgets, we will make $2. If we sell fifty widgets, we will make $100. **This is linear.**

* Example: We crash a car into a wall at 2 mph 50 times, we will probably be OK. If we crash a car into a wall at 100 mph one time, we will probably not be OK. **This is nonlinear.**

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