A Function’s Defining Characteristic

$\blacktriangleright$ Linear Functions

The defining characteristic of a linear function is that it has a constant growth rate.

$$\frac {L(b)-L(a)}{b-a} = m$$

This led to the equation or formula for linear functions: $L(x) = m(x-a) + L(a)$

This tells us that no matter where you are in the domain, of $L$,

• if you move a distance of $1$ in the domain, then the value of $L$ increases by $m$.
  
• if you move a distance of $D$ in the domain, then the value of $L$ increases by $m \cdot D$.
  

Exponential functions do something similar.

$\blacktriangleright$ Exponential Functions

The general template for an exponential function looks like

$$exp(t) = a \cdot b^t \, \text { where } \, a \ne 0 \, \text { and } \, b > 0$$

Their defining characteristic is a constant percentage growth rate: no matter where you are in the domain of an exponential function, if you move the same amount then the function increases by the same percent.

Learning Outcomes

In this section, students will

• analyze exponential functions.
• analyze logarithmic functions.