Shifted Exponential

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almost exponential

Shifted Exponential Functions

Shifted Exponential functions are the sums exponential functions with a nonzero constant term.

They exhibit all of the characteristics of exponential functions, if you take away the constant term.

Shifted Exponential Functions

Shifted exponential functions are exponential functions with a number added on. They can be written in the form

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

where $A$, $B$, $C$, and $D$ are real numbers, $A \ne 0$ and $B \ne 0$ and $D \ne 0$, and $r$ is a positive real number.

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.

Shifted exponential functions can be thought of as compositions our our Core exponential functions with linear functions.

Shifted Exponential Functions

Let $E(k) = r^k $ be a Core exponential function.

Let $L_{in}(t) = B \, t + C$ be a linear function.

Let $L_{out}(y) = A \, y + D$ be a linear function.

General shifted exponential functions are constructed by composing these component functions.

\[ (L_{out} \circ E \circ L_{in})(x) = A \, r^{B \,x + C} + D \]

Here is the graph of $g(x) = 2^x - 3$.

$\log _2(3)$ is the zero of $g(x)$.

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more examples can be found by following this link
More Examples of Elementary Functions

2026-05-29 21:23:01