1.2.1. Exponential Functions

An exponential function has the general form:

𝑦 = 𝑎 ⋅𝑏𝑥

Where: • 𝑎 is the initial value (when 𝑥 = 0), • 𝑏 is the base (growth or decay factor), • 𝑥 is the exponent (often representing time), • 𝑦 is the final amount.

An exponential function helps describe how quantities grow or decay over time. In the formula provided, each component has a specific role:

• The initial value 'a' represents the starting amount before any changes happen. For example, if you're measuring a population, 'a' might be the number of individuals present at the beginning.
• The base 'b' is crucial because it determines how quickly the quantity will change. If 'b' is greater than 1, the quantity grows; if it's between 0 and 1, the quantity decays.
• The variable 'x' usually denotes time, which allows us to see how the quantity changes at different points.
• Finally, 'y' is the result we get after applying the changes, showing what the quantity looks like after a certain time period.

Think of an investment growing in a bank account. If you start with a certain amount of money (that's your 'a'), and your bank gives you interest (that's your 'b'), as time passes (that's your 'x'), you can use this formula to find out how much money you'll have in the end (that's your 'y').

Occurs when a quantity increases by a fixed percentage over regular intervals.

🔹 Formula: 𝑦 = 𝑎(1+𝑟)𝑡

Where: • 𝑎 = initial amount, • 𝑟 = growth rate (as a decimal), • 𝑡 = time, • 𝑦 = amount after time 𝑡.

Exponential growth represents situations where quantities grow quickly over time at a consistent rate. The formula breaks down like this:

• 'a' is the starting value, like how many items or people you have at the beginning.
• 'r' is the growth rate expressed as a decimal, which translates a percentage into a number for mathematical operations. For instance, a 10% growth rate becomes 0.10.
• 't' denotes how many time intervals you’re measuring, helping to quantify how growth accumulates over time periods (years, months, etc.).
• Finally, 'y' tells you the resulting amount after accounting for the growth over the time period you're measuring.

Imagine planting a new tree that grows by 5% each year. If it starts at 2 meters, every year, it gets a little taller based on that growth percentage. If you want to predict how tall it will be after several years, you would use this formula to find out!

Occurs when a quantity decreases by a fixed percentage 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.

In contrast to exponential growth, exponential decay measures how quantities reduce in size over consistent time intervals. Here’s what each part means:

• 'a' starts the process by giving us the initial count or value, just like an object's worth before depreciation.
• 'r' is the decay rate in decimal form, similar to how we convert growth rates. If something depreciates by 20%, 'r' would be 0.20.
• Time 't' is a crucial factor again since it tells us how long we level with this decline.
• Lastly, 'y' is what we’ll have left over time after applying the decay process.

Think about a smartphone battery that loses 10% of its power each hour. You can use the decay formula to find out how much charge remains after a certain time, helping you plan when you might need to recharge.

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