This problem **aims** to understand the **exponential** growth and **exponential** decay.

An **exponential** function is a **function** in which the **exponent** is a variable, and the **base** is positive and $\cancel{=}\space 1$. For **example,** $f(x)=4^x$ is an **exponential** function and the **exponent** is not a mutable but a **specified** constant. $f(x) =x^3$is a **fundamental polynomial** function rather than an **exponential** function. Uninterrupted curved graphs that **never** achieve a **horizontal** asymptote are the **qualities** of exponential functions. Some **practical** phenomena are managed by **logarithmic** or **exponential** functions.

In mathematical **transformation,** exponential growth is a **growth** that grows indefinitely employing an **exponential** function. The **change** that has happened can be either negatively or **positively** executed. The key **assumption** would be that the rate of change is **raising.** When not restrained by **environmental** conditions such as **obtainable** space and sustenance, **populations** of growing **microorganisms,** and certainly any expanding **inhabitants** of any species, may be **expressed** as an exponential growth **function.** The growth of **protection** with compound interest is **another** usage of an **exponential** growth function.

Exponential **decay** happens in mathematical **functions** when the rate by which differences are **happening** is falling and must thus get a **limitation,** which is the exponential **function’s** horizontal **asymptote.** The **asymptote** is the place on the **x-axis** at which the rate of **change matched** near zero. **Exponential** decay may be kept in a **mixture** of techniques. The **decrease** in radioactive **particles** as they fission and decay into **some** other atoms obeys an **exponential** decay curve. A burning item begins to **refrigerate** to a constant **ambient** temperature, or a cold item’s heat will establish an **exponentially** decaying curve. **Exponential** decay may be employed to **define** the discharges of an electric **capacitor.**

The **exponential** growth formula is **employed** to estimate compound interest, find the **population** growth and find the **doubling** time.

**Exponential** growth is **provided** by:

\[f(x)=a(1 +r)x\]

Where $f(x)$ = exponential **growth** function:

$a=$ **Initial** amount,

$r=$ Growth **rate,**

$x=$ Number of time **intervals.**

In exponential growth, the **amount** increases, gradually at first, and then extremely **rapidly.** The pace of **change** increases with **time.**

The **quantity** drops slowly, **observed** by a sharp reduction in the speed of **transition** and rises over time. The **exponential** decay procedure is utilized to **estimate** the lowering in growth. The **exponential** decay procedure can take one of **three** shapes:

\[f(x)=abx\]

\[f(x)=a(1-r)x\]

\[y=y_0e^kt\]

Where,

$a$ or $y_o$ = **Initial** amount,

$b=$ Decay **factor,**

$e=$ **Euler’s** constant,

$r=$ Rate of **decay** (for exponential decay),

$k=$ growth **constant.**

$x$ or) $t$ = time gaps (time can be in days, months, or years, whatever you are **utilizing** should be **uniform** throughout the **situation).**

In **exponential** decay, the amount decreases **initially** very rapidly, and then more gradually. The **pace** of change decreases over the **juncture.** The speed of decay evolves **slower** as time perishes.

## Expert Answer

$y_o$ denotes the **Initial** quantity.

## Numerical Result

In $y=y_oe^kt$ the $y_o$ **represents** the Initial **quantity.**

## Example

In the **decay** function or exponential **growth** $y = y0e^kt$, what does $k$ **represent?**

$k$ represents the **growth** constant.