The defining characteristic of linear functions is that the pairs experience a constant growth rate. If you calculate the rate of change between any two pairs, $(a, f(a))$ and $(b, f(b))$, in the function, you get the same value, $m$.

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

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

Said another way: If you move a fixed amount anywhere in the domain, $\Delta x$, then the function always grows by a proportionally fixed amount, $f(x + \Delta x) - f(x) = m(\Delta x)$. This fixed growth rate is always the same multiple of the change in the domain. That multiple is the constant growth rate.

$\blacktriangleright$ Exponential functions are similar, but it is their percentage growth rate that is constant.

The growth of an exponential function, $g(t)$ over the interval $[a, b]$ is $g(b)-g(a)$. To get a percentage, we compare this back to the starting value, $g(a)$:

$$\frac {g(b)-g(a)}{g(a)}$$

And, then to get the percentage growth rate we average over the interval

$$\frac {\frac {g(b)-g(a)}{g(a)}}{b-a}$$

For an exponential function, this is constant

$$\frac {\frac {g(b)-g(a)}{g(a)}}{b-a} = r$$

or

$$\frac {g(b)-g(a)}{g(a)(b-a)} = r$$

Let’s build such a function up from a given value of $g(0)$.

In general, formulas for exponential functions look like

$$g(t) = g(0) \cdot a^t$$

Exponential Functions

Said another way: If you move a fixed amount in the domain, $\Delta x$, then the change in an exponential function, is a fixed multiple of the function value.

$$k \cdot r^{x + \Delta x} - k \cdot r^x = k \cdot r^x (r^{\Delta x} - 1)$$

• Linear growth means a fixed multiple of the change in domain value. It is independent of the function value.
• Exponential growth means a fixed multiple of the function value.

There are two types of exponential functions corresponding to $0<r<1$ or $1<r$.

• If $0<r<1$, then greater positive exponents make the function value smaller.
  In the other direction, a negative exponent essentially gives us the reciprocal of $r$, which would be greater than $1$ here. Therefore, greater negative exponents result in bigger function values.
  
• If $1<r$, then greater positive exponents make the function value bigger.
  In the other direction, a negative exponent essentially gives us the reciprocal of $r$, which would be less than $1$ here. Therefore, greater negative exponents result in smaller postive function values.

The graphs of $y = Y(x) = 3 \cdot \left (\frac {1}{2}\right )^x$ and $z = W(t) = 3 \cdot 2^t$ are shown below.

There is no vertical asymptote. The domain of exponential functions is all real numbers. $y=0$ is a horizontal asymptote on both graphs. The sign of an exponential function is given by the coefficient.

Since these formulas are centered around the exponent, they follow the exponent rules:

• $a^n \cdot a^m = a^{n+m}$
• $\frac {a^n}{a^m} = a^{n-m}$
• $(a^n)^m = a^{n \cdot m}$
• $a^n \cdot b^n = (a \cdot b)^n$
• $\frac {a^n}{b^n} = \left (\frac {a}{b}\right )^n$

Note: In the template for exponential functions, There is a leading coefficient for the function and there is a leading coefficient for the linear function inside the exponent.

This is the most general form of an exponential function. It is equivalent to the basic form, meaning it can be transformed into the basic form using our exponent rules.

$$f(x) = A \cdot r^{B \, x + C}$$

$$f(x) = A \cdot r^{B \, x} \cdot r^C$$

$$f(x) = A \cdot (r^{B})^x \cdot r^C$$

$$f(x) = A \cdot r^C \cdot (r^{B})^x = a \cdot R^x$$

$A \cdot r^C$ is the new leading coefficient.

$r^{B}$ is the new base.

If we have the characteristics of this exponential function memorized, then we can compare other exponential functions back to this one.

Exponential functions are those functions that CAN be described with a formula like

$$A \cdot r^x$$

Using exponent rules, we can rewrite any exponential formula into this form. Therefore, it is nice to memorize the characteristics of one of these basic forms and then compare all others back to it.

Even if $r<1$, the base can be rewritten as $r = (r^{-1})^{-1}$. The first $-1$ exponent stays with the $r$, to make $r^{-1}= \frac {1}{r} > 1$. The second $-1$ moves up to the exponent and changes the sign of the exponent.

Example:

$$\left ( \frac {1}{3} \right ) = \left ( \left ( \frac {1}{3} \right )^{-1} \right )^{-1} = ( 3 )^{-1}$$

$$2 \left ( \frac {1}{3} \right )^{5-6x} = 2 \left ( ( 3 )^{-1} \right )^{5-6x} = 2 (3)^{(-1)(5-6x)} = 2 (3)^{-5+6x}$$

Shifted Exponential Functions

Note: In the template for shifted exponential functions, There is a leading coefficient for the function and there is a leading coefficient for the linear function inside the exponent.

The main different between shifted exponential and exponential functions is that while exponential functions do not have zeros, a shifted exponential function may have a zero.