Exponential Growth

Today we'll dive into exponential functions. Can anyone tell me what an exponential function is?

Is it like a function that increases really fast?

Great observation! An exponential function indeed grows rapidly. It follows the formula 𝑦 = 𝑎 ⋅ 𝑏^𝑥. Here, 𝑎 is our starting value, and 𝑏 is the base which determines if it grows or decays.

If 𝑏 is greater than 1, does it mean growth?

Exactly! If 𝑏 > 1, we have exponential growth. This contrasts with decay when 0 < 𝑏 < 1.

Can we see a visual representation of this?

Of course! The graphs are curved. They never touch the x-axis as they approach it. This concept is vital in various fields. Can anyone think of a real-life example where exponential growth is observed?

Like population growth?

Exactly! Excellent example! Let's summarize: exponential functions model rapid growth when 𝑏 > 1.

Next, let's explore the exponential growth formula, which is 𝑦 = 𝑎(1+𝑟)^𝑡. Can someone break down this formula?

𝑎 is the initial amount, right?

Correct! And 𝑟 represents the growth rate. What about 𝑡?

Isn't it the time over which we measure growth?

Exactly! So, if we want to calculate how a quantity grows over time, we input values for 𝑎, 𝑟, and 𝑡 into this formula.

Can you show us an example?

Sure! If a population is 500 and doubles every 3 hours, what is its population after 9 hours? We have 𝑎 = 500, 𝑟 = 1, and 𝑡 = 3 doubling periods. It grows to 4000.

That's amazing! It really shows how fast exponential growth can happen.

Exactly! Let's recap: the formula for exponential growth is 𝑦 = 𝑎(1+𝑟)^𝑡, which shows how quantities increase over time.

Now let's shift focus to exponential decay. Can anyone describe what it is?

Is it when things get smaller over time?

Exactly! In exponential decay, quantities decrease at a steady percentage over time, so we use 𝑦 = 𝑎(1−𝑟)^𝑡. What do 𝑎, 𝑟, and 𝑡 stand for?

Initial amount, decay rate, and time!

Spot on! A common example is how a car depreciates. If a $20,000 car loses 15% of its value each year, we can find its value after 5 years.

How do we do that?

We would input 𝑎 = 20000, 𝑟 = 0.15, and 𝑡 = 5 into the formula. We find it's worth about $8,874.

That's interesting! I didn’t realize depreciation works this way.

It's very relevant in finance and economics. To sum up, exponential decay occurs based on 𝑦 = 𝑎(1−𝑟)^𝑡, leading to significant decreases over time.