From what we can see in Exploration $1,$ we know that there are some specific groups of exponential functions that increase by a constant factor or percentage over equal intervals of $x .$ For these cases, we say that the functions exhibit exponential growth.

Let's recall the form of an exponential function.

y = a b^{x}

If we consider

a > 0,

the exponential function will be increasing over time only if

b > 1 .

Therefore, we can rewrite the exponential function as shown below.

y = a b^{x} \Leftrightarrow y = a (1 + r)^{x}

Here

r

represents how fast the function is growing as a percentage, written in decimal form. This is why

r

is known as the growth rate in these cases. On the other hand, if the function is decreasing by a constant factor or percentage over equal intervals of

x,

we say that it exhibits exponential decay.

For this to happen, we need that

0 < b < 1 .

If this is the case, we can rewrite the exponential function as shown below.

y = a b^{x} \Leftrightarrow y = a (1 - r)^{x}

Here

r

represents how fast the function is decreasing as a percentage, written in decimal form. This is why

r

is known as the decay rate in these cases.

Exercise