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Mathematical Expression Editor
exponential form
The template for the formula of the basic exponential function looks like
\[ A \cdot r^x \, \text { with } \, A, r \in \mathbb {R} \, | \, A \ne 0, \, r > 0 \]
Exponential functions are those functions that CAN be represented by a formula of
the form \(A \cdot r^x\).
Exponential Functions
\(f(x) = 7 \cdot 2^{3x+1} \) represents an exponential function, because this formula can be rewritten in the
form \(a \cdot r^x\).
\[ f(x) = 7 \cdot 2^{3x+1} \]
\[ f(x) = 7 \cdot 2^{3x} \, 2^1 \]
\[ f(x) = 7 \cdot 2 \, 2^{3x} \]
\[ f(x) = 7 \cdot 2 \, (2^3)^x \]
\[ f(x) = 14 \cdot 8^x \]
Exponential Functions
\[ g(t) = \frac {5 \cdot 3^{8 t + 9}}{7 \cdot 3^{6 t + 4}} \]
represents an exponential function, because this formula can be rewritten in the form
\(a \cdot r^t\).
\[ g(t) = \frac {5 \cdot 3^{8 t + 9}}{7 \cdot 3^{6 t + 4}} \]
\[ g(t) = \frac {5}{7} \cdot \frac {3^{8 t + 9}}{3^{6 t + 4}} \]
\[ g(t) = \frac {5}{7} \cdot 3^{(8 t + 9)-(6 t + 4)} \]
The template for the formula of the basic shifted exponential function looks
like
\[ A \cdot r^x + D \, \text { with } \, A, D, r \in \mathbb {R} \, | \, A \ne 0, \, r > 0 \]
However, this is not how we usually encounter exponential and shifted exponential
functions.
We usually encounter exponential and shifted exponential functions in a more general
form.
\[ A \cdot r^{B \, x + C} \]
\[ A \cdot r^{B \, x + C} + D \]
\(r\) is called the base. \(A\) is called the leading coefficient of the function. \(B\) is the leading coefficient of the linear function in the exponent. \(D\) is called the constant term.