1. Exponential Growth and Decay

Today, we'll explore exponential functions, which have a specific form denoted as $y = a \cdot b^x$. Has anyone encountered this form before?

I've seen it in my previous class, but I'm not sure exactly what it means.

That's okay! Let's break it down. Here, $a$ represents the initial value when $x = 0$, and $b$ is the growth or decay factor. Can anyone tell me what happens if $b > 1$?

It means the function is growing!

Correct! Now, if $0 < b < 1$, what would that indicate?

It would indicate decay.

Exactly! We will delve deeper into both growth and decay.

Let's discuss exponential growth. The formula for it is $y = a(1 + r)^t$. Can someone explain what $r$ stands for?

Isn't $r$ the growth rate in decimal form?

Correct! Great job! Now, consider this example: if a population of 500 bacteria doubles every 3 hours, what can we calculate after 9 hours?

We can find out how many doubling periods are in 9 hours!

That's 3 periods, right?

Yes, so we calculate $y = 500 \cdot 2^3 = 4000$ bacteria. Excellent!

Switching gears, let's look at exponential decay, represented by $y = a(1 - r)^t$. Who can remind me what this formula calculates?

It calculates how much a quantity decreases by a certain rate over time.

Absolutely! Let's think about a car worth $20,000 that depreciates at 15% each year. Can someone use the formula to find its value after 5 years?

I think $a = 20000, r = 0.15, t = 5$. So, $y = 20000(0.85)^5$.

Great! And what do you get when you calculate that?

About $8,874!

Lastly, how do we graph exponential functions? What does the graph look like?

It’s a curve that never touches the x-axis!

Right! It approaches the x-axis but never crosses it. This characteristic is crucial for understanding limits in functions.

So, no matter how long we wait, it never actually reaches zero?

Exactly! Whether it’s growth or decay, the behavior of the graph remains significant.