1.2. Key Concepts

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 describes how quantities change over time based on their current value. The equation consists of several parts: 𝑎, which is the starting amount; 𝑏, the base which indicates growth or decay; 𝑥, which generally represents time; and 𝑦, the result after applying the function. This format lets us model various real-world phenomena like population growth or financial interest.

Imagine you have a savings account that earns interest. The amount of money you have in the account grows based on both your initial deposit (the initial value, 𝑎) and the interest rate (the growth factor, 𝑏). Over time (represented by 𝑥), your account balance (𝑦) will increase exponentially, especially if you keep adding money.

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 is characterized by a constant percentage increase over time. The formula allows you to calculate the future value after a certain period by considering the initial amount and the growth rate. This means that as time passes, not only the initial amount grows, but the increase itself keeps getting larger.

Think of a small plant that grows. If it grows by 10% each week, after several weeks, it won't just be a little bigger; it will be much taller because the growth compounds. Each week's growth adds a larger amount than the previous week as it gets taller.

✅ Example 1: A population of 500 bacteria doubles every 3 hours. What is the population after 9 hours?

Solution: • Initial population 𝑎 = 500, • Growth rate 𝑟 = 100% = 1, • Time 𝑡 = 9/3 = 3 doubling periods. 𝑦 = 500(2)3 = 500×8 = 4000 Answer: 4000 bacteria

In this example, the initial population of bacteria is 500, and they double every 3 hours. To find the population after 9 hours, we determine that 9 hours equals three 3-hour periods. We then apply the growth formula, where the bacteria population is calculated as 500 multiplied by 2 raised to the power of the number of doubling periods (3). Hence, 500 multiplied by 8 (which comes from 2 raised to the power of 3) equals 4000.

Consider how fast a viral video can spread. If a video is watched by 500 people, and every three hours the number of viewers doubles, in just 9 hours, the view count can spike dramatically to about 4000. This showcases how rapidly information can circulate through a network.

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.

Exponential decay refers to a situation in which a quantity decreases at a fixed percentage rate over time. The formula incorporates the initial amount and the decay rate to provide the quantity remaining after a given period. Unlike linear decreases, the reduction accelerates as time goes on because each decrease is calculated from a smaller and smaller current value.

Imagine a car's value depreciating as it ages. If a car starts at $20,000 and loses 15% of its value annually, with each passing year, its resale value diminishes more significantly, not just because it’s old, but because the 15% loss applies to an already reduced amount.

✅ Example 2: A car worth $20,000 depreciates at a rate of 15% per year. What will it be worth after 5 years?

Solution: • 𝑎 = 20,000, • 𝑟 = 0.15, • 𝑡 = 5 𝑦 = 20000(1−0.15)5 = 20000(0.85)5 ≈ 20000×0.4437 = 8874 Answer: Approx. $8,874

Here, the initial value of the car is $20,000, and it's losing 15$8,874 after 5 years, showing the impact of exponential decay on its worth.

Think about how electronics, like smartphones, lose value the moment they're sold. If a new phone starts at $1000 and depreciates at a specific rate every year, by the time you decide to sell it after a few years, it could be worth significantly less, just like the car example.

• 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.

This section highlights key attributes of exponential functions. Growth occurs when the base (𝑏) is greater than one, indicating an increase, while decay is apparent when the base is between 0 and 1. The graphical representation differs significantly between growth (which rises sharply) and decay (which gradually levels off but approaches zero without ever actually reaching it).

Think of a balloon. If you keep blowing it up, it’ll get larger rapidly (representative of growth). Conversely, if you start releasing air, it gradually gets smaller and smaller but won’t disappear instantly, illustrating decay.

• The graph of an exponential function is a curve. • It passes through the point (0,𝑎). • The x-axis (horizontal) is often a time axis. • The y-axis shows quantity (population, value, etc.). • The function never crosses the x-axis; it gets very close (asymptote).

When graphing an exponential function, the output forms a distinct curve. It begins at point (0,𝑎) and stretches along the x-axis, with time represented on the horizontal axis and the amount on the vertical axis. Notably, the graph never intersects the x-axis, which represents an asymptote; it only approaches it. This characteristic is crucial in understanding that quantity might decrease to a very small number but never actually become zero in practical applications.

Consider the trend of a stock price. It may drop gradually without ever hitting zero; it just seems to get closer and closer to some minimum value. The curve you get from plotting this behavior demonstrates this characteristic, reflecting the volatility inherent in market-based changes.

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