Shape of the Graph of an Exponential Function where $0 < a < 1$

y-intercept of the Graph of an Exponential Function

Point $(1, a)$ on the Graph of an Exponential Function

Point $(-1, \frac{1}{a})$ on the Graph of an Exponential Function

Domain of an Exponential Function

Range of an Exponential Function

Example: Graphing f(x) = \left(\frac{1}{2} ight)^x and g(x) = \left(\frac{1}{3} ight)^x

Requirement that the Base of an Exponential Function is Positive ($a > 0$)

Shape of the Graph of an Exponential Function where $a > 1$

One-to-One Property of Exponential Functions

Horizontal Asymptote of the Graph of an Exponential Function

x-intercept of the Graph of an Exponential Function

Horizontal Translation of an Exponential Function

Vertical Translation of an Exponential Function

Natural Base $e$

Requirement that the Base of an Exponential Function is Not Equal to 1

Example: Graphing $f(x) = 2^x$ and $g(x) = 3^x$

Example: Graphing $f(x) = 4^x$

Example: Graphing f(x) = \left(\frac{1}{4} ight)^x

Example: Graphing g(x) = \left(\frac{1}{5} ight)^x

Applications of Exponential Functions

Difference Between Exponential and Polynomial Functions

As a financial analyst trainee reviewing investment growth models, you encounter various mathematical formulas. Compound interest investments are modeled using an exponential function of the form $f(x) = a^x$. Based on the formal definition of an exponential function, which of the following statements correctly identifies its required mathematical characteristics?

As a financial analyst trainee, you are reviewing various mathematical models used to project growth. Match each component or characteristic of the exponential function $f(x) = a^x$ with its correct description according to its formal mathematical definition.

A market researcher is using the mathematical model $f(x) = a^x$ to project the rapid growth of a new product's user base. To correctly classify this model as an exponential function, the researcher must identify that the variable $x$ is located in the ____.

Constraints on the Base of an Exponential Function

A demographics clerk is using the mathematical model $f(x) = a^x$ to project city population trends. According to the formal definition of an exponential function, the constant base $a$ is permitted to be any positive real number.