# Exponential Functions

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## Exponential Function

Many functions containing a variable **exponent**, are called exponential functions. Formally, any function that can be written in the following form is an exponential function.

$y=a \cdot b^x$

Here, the coefficient, $a,$ is the $y$-intercept, which is sometimes referred to as the initial value. The base, $b,$ can be interpreted as the constant multiplier. To ensure that $y$ is an exponential function, there are restrictions on $a$ and $b.$

### $a\neq 0$

If the coefficient $a$ is $0,$ the function becomes
$y=0\cdot b^x \quad \Rightarrow \quad y=0.$
This is a line along $y=0,$ and thus, a linear relationship. Therefore, if $a=0$ the function is **not** exponential.

### $b \gt 0$ and $b \neq 1$

## Graphing an Exponential Function using the Function Rule

## Exponential Growth

Exponential growth occurs when a quantity increases by the same factor over equal intervals of time. This leads to an exponential function, where the independent variable in the exponent, $t,$ is time.
$y = a \cdot b^t$
Since the quantity **increases** over time, the constant multiplier $b$ has to be **greater** than $1.$ Thus, $b$ can be split into two terms, $1$ and $r,$ where $r$ is some positive number. The resulting function is called an *exponential growth function*.

The constant $r$ can then be interpreted as the *rate of growth*, in decimal form. A value of $0.06,$ for instance, means that the quantity increases by $6\,\%$ over every unit of time. As is the case with all exponential functions, $a$ is the $y$-coordinate of the $y$-intercept. The base of the power, $1 + r,$ is known as the *growth factor*, and since it's greater than $1$ the quantity grows faster and faster, without bound.

## Exponential Decay

Exponential decay is very similar to exponential growth. However, instead of the quantity increasing, it **decreases** by the same factor over equal segments of time. This behavior is also described by an exponential function,
$y = a \cdot b^t.$
However, the constant multiplier $b$ is now less than $1.$ Since it is less than $1,$ it can be expressed as a subtraction of $r$ from $1,$ where $r$ is some positive number less than $1.$ The resulting function is called an *exponential decay function*.
$y = a \cdot (1 - r)^t$
In this context, the constant $r$ can be interpreted as the *rate of decay*, in decimal form. A value of $0.12,$ as an example, means that the quantity decreases by $12\,\%$ over every unit of time. Same as for an exponential growth function, $a$ is the $y$-coordinate of the $y$-intercept. However, since the base $1 - r$ is smaller than $1,$ the quantity decays towards $0$ over time.

## Compound Interest

When money is deposited to a savings account, interest is accrued, often yearly. Different types of interest work in different ways. When the interest earned is then added to the original amount, future interest accrues for a larger amount. This is called compound interest. To calculate the balance on the account at a specific time, an exponential growth function can be used. When the interest is compounded yearly, the balance can be modeled with a function.

$y = P(1 + r)^t$

In this context, $P$ stands for the *principal*, which is the initial amount of money, and $r$ is the interest rate in decimal form. If the interest is not compounded yearly, the function looks a little different.

$y = P \left( 1 + \dfrac r n \right)^{nt}$

The constant $n$ is the number of times the interest is compounded per year, while $r$ is still the annual interest rate. For an account with the principal $$100$ and an annual interest of $15\,\%$ compounded twice a year, the growth function is:

$B(t) = 100\left( 1 + \dfrac{0.15}{2} \right)^{2t}$

Notice that this function grows continuously, whereas, in reality, the account balance only increases at the times of compound. Graphing the function together with the actual balance of the account will highlight how it can be used in practice.

Every time the interest is compounded, in this case every half year, the value of $B$ is equal to the account balance. However, at all other times, it is not. To find, for instance, the account balance after $1.75$ years, $B(1.5)$ should then be evaluated, since that was the last time interest compounded.## Exercises

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