Shifted Exponential

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charcateristics

Shifted Exponential Functions

Shifted Exponential functions are sums of exponential functions and constant functions.

They are not exponential functions, because constant percentage growth doesn’t work with an added constant.

However, as far as function properties go, they are quite similar.

We can account for all of the leading coefficients and added constants through composition with linear functions.

Our general template for shifted exponential functions looks like

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

Shifted exponential functions can be viewed as compositions.

Let $CE(t)=r^t$ be a Core exponential function.
Let $L_{in}(u) = B \, u + C$ be a linear function.
Let $L_{out}(v) = A \, v + D$ be a linear function.

Then, our general shifted exponential function can be expressed as a composition.

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

Now we can use the Chain Rule to establish whether a shifted exponential function increases or decreases.

Shifted Exponential Function

Here is the graph of $g(x) = \left (\frac {1}{3}\right )^x + 2$.

$g$ is the sum of an exponential function, $\left (\frac {1}{3}\right )^x$, and the constant function $2$.

The exponential function values have all been increased by $2$.

Graphically, the horizontal asymptote in the graph has shifted vertically by 2 units to $y=2$.

Behavior:

We can view $g(x)$ as a composition.

Let $CE(t) = \left (\frac {1}{3}\right )^t$, a decreasing core exponential function.
Let $L_{out}(v) = v + 2$, an increasing linear function.

Then, $g(x)$ can be expressed as a composition.

\[ g(x) = \left (\frac {1}{3}\right )^x + 2 = (L_{out} \circ CE)(x) \]

Now we can use the Chain Rule to establish whether $g(x)$ increases or decreases.

\[ inc \circ dec = dec \]

This agrees with the graph.

Exponential functions are functions that exhibit a constant percentage growth rate. There is some constant $p$, such that

\[ f(x+1) = p \cdot f(x) \]

All exponential functions CAN be written in the form $f(x) = p^{a \, x + b}$. All exponential functions are a number raised to a linear function.

Note: Since $p = e^{ln(p)}$, every exponential function CAN be written in the form $e^{a \, x + b}$. So, we really only need study base $e$ exponential functions.

$\blacktriangleright $ Shifted exponential functions

$y(x) = \left (\frac {1}{3}\right )^x + 2$ cannot be written in the form $f(x) = p^{a \, x + b}$.

The added constant term prevents $y(x)$ from being written in our exponential form. It is not an exponential function.

It is a shifted exponential form.

However, for our purposes shited exponential functions fits nicely into our composition story. So, shifted exponential functions are a part of the exponential story, just another chapter.

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

Note: In Calculus, we will see that when you write exponential formulas with base $e$, then nice things happen with the calculations. So, we like base $e$.

Shifted Exponential Analysis

Domain: The domain of every shifted exponential function is $(-\infty , \infty )$.

Zeros: If a shifted exponential function has a zero, then it only has one.

Continuity: Shifted exponential functions are continuous.

End-Behavior: In one direction, shifted exponential functions become unbounded. In the other direction, exponential functions tend to the constant term.

When the base is greater than $1$,

In the domain, when you move in the direction that makes the exponent unbounded negative, the function tends to the constant term.
In the domain, when you move in the direction that makes the exponent unbounded positive, the function is unbounded. Its sign is given by the leading coefficient.

When the base is less than $1$,

In the domain, when you move in the direction that makes the exponent unbounded negative, the function is unbounded. Its sign is given by the leading coefficient.
In the domain, when you move in the direction that makes the exponent unbounded positive, the function tends to the constant term.

Behavior (Increasing and Decreasing): Shifted exponential functions are either increasing or they are decreasing.

The Chain Rule can tell you which.

Global Maximum and Minimum: Shifted exponential functions do not have a global maximum or global minimum.

Local Maximums and Minimums: Shifted exponential functions do not have local maximums or minimums.

Range: The range of a shifted exponential function is either $(-\infty , D)$ or it is $(D, \infty $.

Reading Coefficients

Behavior:

Shifted exponential functions behave similarly to exponential functions.

Their behavior can be identified through the Chain Rule.

If we write shifted exponential functions in terms of base $e$, then we can compare leading coefficients for behavior.

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

$A$ is the leading coefficent for the function.
$B$ is the leading coefficent of the exponent.

Comparing these back to our basic exponential functions, we get

$A > 0$ and $B > 0$ gives an increasing shifted exponential function.
$A < 0$ and $B > 0$ gives a decreasing shifted exponential function.
$A > 0$ and $B < 0$ gives a decreasing eshifted xponential function.
$A < 0$ and $B < 0$ gives an increasing shifted exponential function.

$\blacktriangleright $ When the leading coefficients are the same sign, then the shifted exponential function is increasing.

$\blacktriangleright $ When the leading coefficients are different signs, then the shifted exponential function is decreasing.

End-Behavior:

The big difference between exponential functions and shifted exponential functions is the end-behavior.

While exponential functions tend to $0$ in one tail of the domain, shifted exponential functions tend to the added constant value.

$shexp(x) = A \cdot e^{B \, x + C} + D$ will tend to $D$ in the tail where the exponent is negative.

$shexp(x) = A \cdot e^{B \, x + C} + D$ will become unbounded in the tail where the exponent is positive. The sign will be given by the leading coefficient, $A$.

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2026-05-30 01:34:17