Solutions (Selected)

Today, we'll dive into exponential functions, which are crucial for modeling change in populations and values over time. Who can tell me the general form of an exponential function?

Isn't it something like 𝑦 = 𝑎 ⋅ 𝑏𝑥?

Exactly! In this formula, 𝑎 represents the initial value, and 𝑏 is the growth factor. Remember, if 𝑏 > 1, it's growth; if 0 < 𝑏 < 1, it's decay. A mnemonic to remember this is 'Growth is greater than 1, decay is less.'

Why is it essential to understand the difference between growth and decay?

Great question! Understanding this helps us analyze real-world situations, like populations or finances. Now, let’s apply this to a practical example.

Let’s say we have a population of 1,000 people, which grows at a rate of 4% per year. How would we calculate the population in 10 years?

We would use the growth formula: 𝑦 = 𝑎(1 + 𝑟)𝑡, right?

Right! We have 𝑎 = 1000, 𝑟 = 0.04, and 𝑡 = 10. Can someone calculate 𝑦 for me?

So, 𝑦 = 1000 * (1.04)^10, which approximately equals 1480.2.

Spot on! The population will be about 1,480 people after 10 years. Remember, this shows how quickly populations can grow exponentially!

Now let’s look at exponential decay. A car worth $20,000 depreciates by 15% each year. Using the decay formula, how would we find its value after 5 years?

We should use the formula 𝑦 = 𝑎(1 - 𝑟)𝑡.

Correct! What do we have for our values?

Here, 𝑎 = 20000, 𝑟 = 0.15, and 𝑡 = 5.

Can someone calculate the final value?

So, 𝑦 = 20000 * (0.85)^5, which is about $8,874.

Great job! This illustrates the importance of understanding depreciation.