# How do you identify equations as exponential growth, exponential decay, linear growth or linear decay y=(3/4)^3?

Jul 28, 2015

If the function is $y = f \left(x\right) = {\left(\frac{3}{4}\right)}^{x}$, then it's exponential decay.

#### Explanation:

Functions of the form $y = f \left(x\right) = a \cdot {b}^{x}$, where $a > 0$, $b > 0$, and $b \ne 1$ are called exponential functions. If $0 < b < 1$, then it's a decreasing function and is called an exponential decay function. If $b > 1$, then it's an increasing function and is called an exponential growth function.

Exponential decay functions have the property that each 1 unit increase in $x$ results in a decrease of $y$ by $100 \cdot \left(1 - b\right)$ percent. For example, if $b = 0.96$ and $x$ increases by 1 unit, then $y$ goes down by 4%. Here are some algebraic details that confirm this: $\frac{\Delta y}{y} = \frac{a \cdot {b}^{x + \Delta x}}{a \cdot {b}^{x}} = \frac{\cancel{a} \cdot \cancel{{b}^{x}} \cdot {b}^{\Delta x}}{\cancel{a} \cdot \cancel{{b}^{x}}} = {b}^{\Delta x} = {b}^{1} = 0.96$, a 4% decrease.

On the other hand, exponential growth functions have the property that each 1 unit increase in $x$ results in an increase of $y$ by $100 \cdot \left(b - 1\right)$ percent. For example, if $b = 1.08$ and $x$ increases by 1 unit, then $y$ goes up by 8%. Here are some algebraic details that confirm this: $\frac{\Delta y}{y} = \frac{a \cdot {b}^{x + \Delta x}}{a \cdot {b}^{x}} = \frac{\cancel{a} \cdot \cancel{{b}^{x}} \cdot {b}^{\Delta x}}{\cancel{a} \cdot \cancel{{b}^{x}}} = {b}^{\Delta x} = {b}^{1} = 1.08$, an 8% increase.

You can watch my "Quick Precalc Review" videos to see a lot of examples like this (I start by talking about linear functions but also get to exponential functions by Video #7).