Função linear vs função exponencial

Exponential and linear refer to the type of a function by looking at the power of the independent variable.

A linear function is one where the independent variable is to the power of $1$. For example, in the linear equation $y=mx+b$, $x$ is the aforementioned independent variable. The term linear comes from the plot of the function; regardless of the values of $m$ and $b$, the graphed function will always be a line.

An exponential function is one where the independent variable is to a non-trivial (not $0^{\text{th}}$ or $1^{\text{st}}$) power. These are typically of the form $y=a\cdot b^x$. The term exponential comes from the use of exponentiation in the independent variable.

The graph below shows a linear function (red) and an exponential function (blue).

One of the most important distinctions between linear and exponential functions is how (and how quickly) they increase or decrease. Linear functions increase proportionally; an increase in $x$ has a corresponding additive increase in $y$. Exponential functions, however, increase exponentially; that is, an increase in $x$ has a corresponding multiplicative increase in $y$.

(Video) Linear vs. Exponential Functions

by Mathematics Exemplified

Here's a good video by Mathematics Exemplified showing you how to determine if a set of points is a linear line or an exponential line. He also shows you how to find the equation for the line given the points.

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To do the method he describes you have to make sure your $x$ values are increasing at a constant rate. That is, they change by the same number each time.

Next, look at the $y$ values. If the $y$ values are also increasing at a constant rate then your function is linear. In other words, a function is linear if the difference between terms is the same.

For exponential functions the difference between terms will not be the same. However, the ratio of terms is equal.

(Video) Linear vs Exponential

by Sinclair Johnston

Here's a great video detailing the differences between linear and exponential growth!

$\large{\text{Summary}}$

Linear functions grow by adding or subtracting while exponential functions growth by multiplying or dividing.