1.2.2.1. Formula

🔹 Formula: $y = a(1 + r)^t$ Where: • 𝑎 = initial amount, • 𝑟 = growth rate (as a decimal), • 𝑡 = time, • 𝑦 = amount after time 𝑡.

The formula for exponential growth helps calculate how much a quantity will increase over time at a constant rate. Let's break it down:

• $a$ is the initial amount or starting value before any growth happens.
• $r$ represents the growth rate expressed as a decimal. For example, if the growth rate is 4%, we would write this as 0.04.
• $t$ stands for time, indicating how many time intervals (like years or months) we are considering for the growth.
• Finally, $y$ is the final amount after the specified time period has passed. So, if you plug in the values of $a$, $r$, and $t$ into this formula, you can find out how much the quantity has grown over time.

Think of planting a tree. The initial height of the tree (like $a$) is how tall it is when you first plant it. Every year, it grows at a certain rate (like $r$), say it grows 5% taller each year. After a few years (specified by $t$), you can calculate how tall your tree will be using this formula. This helps illustrate how initial conditions and rates can accumulate over time.

🔹 Formula: $y = a(1 - r)^t$ Where: • 𝑎 = initial amount, • 𝑟 = decay rate (as a decimal), • 𝑡 = time, • 𝑦 = amount after time 𝑡.

The exponential decay formula calculates how much a quantity will decrease over time at a constant rate. Here's how it works:

• Again, $a$ is where we start; it represents the initial amount before any decay has occurred.
• $r$ is the decay rate in decimal form. If something decreases by 15%, you would express this as 0.15.
• $t$ represents the time period over which the decay happens.
• $y$ is the final amount left after decay takes place for that time duration. You can use this formula to see how much of something, like a car’s value or radioactive material, remains after a certain period.

Consider a phone battery that starts off fully charged at 100%. If it loses 20% of its charge every hour, you can think of its initial charge ($a$) as 100%. Each hour, you would apply the decay rate (20% or 0.20) to find out how much charge is left. Over time, you can use the decay formula to see how the battery charge decreases until it potentially runs out.

🧮 Important Points • If 𝑏 > 1, it’s exponential growth. • If 0 < 𝑏 < 1, it’s exponential decay. • Exponential growth graphs increase rapidly. • Exponential decay graphs decrease and flatten but never hit zero.

The key characteristics differentiate between exponential growth and decay:

• If the base $b$ of the exponential function (which influences growth and decay) is greater than 1, the function describes exponential growth, indicating that the quantity is increasing rapidly.
• Conversely, if $b$ is between 0 and 1, it represents exponential decay, meaning the quantity is decreasing over time.
• On the graph, exponential growth shows a steep upward curve, while exponential decay starts high and flattens out as it approaches zero but never actually reaches it, creating what is called an asymptote.

Think of a city that is growing rapidly due to a tech boom, where every year more and more people move in (exponential growth). The city's population graph would soar steeply upwards. In contrast, think of a building that is being demolished slowly. Each day it loses a bit of its height; the graph would show a gradual decline but never completely disappear, indicating it will always have some remnant until it is fully taken down.

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