Lesson 19.1: Exponential Functions

Property

An exponential function has the form

The constant $a$ is the $y$-intercept of the graph because $f(0) = a \cdot b^0 = a \cdot 1 = a$.
The positive constant $b$ is called the base. We do not allow $b$ to be negative, because if $b < 0$, then $b^x$ is not a real number for some values of $x$. We also exclude $b = 1$ because $1^x = 1$ for all values of $x$, which is a constant function.

Examples

• The function $f(x) = 5(2)^x$ is an exponential function where the initial value is $a=5$ and the growth factor is the base $b=2$.

• The function $P(t) = 100(0.75)^t$ represents exponential decay with an initial amount of $100$ and a decay factor of $0.75$.

Property

Power of a Power Rule: $(a^b)^c = a^{bc}$

Product of Powers Rule: $(a^b)(a^c) = a^{b+c}$

Property

For an exponential function $f(x) = b^x$:

• The graph always passes through the point $(0, 1)$, which is the y-intercept.
• The x-axis ($y=0$) is a horizontal asymptote, meaning the graph gets infinitely close but never touches it.
• If the base $b > 1$, the function is always increasing (representing exponential growth).
• If $0 < b < 1$, the function is always decreasing (representing exponential decay).

Examples

• The graph of $f(x) = 3^x$ is an increasing function. It passes through $(0, 1)$ and rises sharply to the right as $x$ increases.