1.8. Chapter Summary

Exponential 𝑦 = 𝑎(1+𝑟)𝑡 Increase by constant rate (%) over time

Exponential growth is represented by the formula 𝑦 = 𝑎(1+𝑟)𝑡. Here, 𝑦 is the final amount after time 𝑡, 𝑎 is the initial amount, and 𝑟 is the growth rate expressed as a decimal. This formula implies that the quantity increases by a specific percentage of its current value at each time interval.

Imagine a small investment growing in a bank due to compound interest. If you invest $100 with an interest rate of 5$100, but also on the interest that accumulates. So, after one year, you will have $105, and in the next year, you will earn interest on$105, leading to even bigger growth.

Exponential 𝑦 = 𝑎(1−𝑟)𝑡 Decrease by constant rate (%) over time

The exponential decay is modeled by the formula 𝑦 = 𝑎(1−𝑟)𝑡. In this case, 𝑎 represents the initial amount, 𝑟 is the decay rate as a decimal, and 𝑡 signifies time. Each time period, the remaining quantity decreases by a fixed percentage of its current value.

Consider a smartphone battery. If a battery starts at 100% and loses 20% of its charge each hour, it doesn't lose 20% of the original 100% each hour. After one hour, it would have 80%, and in the second hour it would lose 20% of that 80%, resulting in a continually diminishing charge.

Base 𝑏 If 𝑏 > 1: Growth; If 0 < 𝑏 < 1: Decay

The base (b) of an exponential function indicates the nature of the growth or decay. If b is greater than 1, it indicates exponential growth, meaning the quantity is increasing. Conversely, if b is between 0 and 1, the function represents exponential decay, indicating that the quantity is decreasing over time.

Think of b like the flavor of a recipe. When making a cake, if you increase the amount of sugar (b > 1) it becomes sweeter over time. However, if you reduce the sugar (b < 1), the cake becomes less sweet, representing decay.

Graph Shape Curved, not linear Approaches zero but never touches it

The graph of an exponential function has a characteristic curve. For exponential growth, it rises steeply as time progresses. For exponential decay, the graph decreases but approaches the x-axis without ever touching it. This illustrates how quantities can diminish but never fully reach zero, forever approaching it asymptotically.

Imagine a candle burning. The candle’s wax diminishes over time but never fully vanishes until completely burnt. The rate may slow down, which reflects how the graph approaches zero but never actually reaches it.

Applications Finance, biology, physics Real-world modeling

Exponential functions are widely used in real-world scenarios. In finance, they model growth through interest on savings or loans. In biology, they can explain population dynamics. In physics, they help describe decay processes such as radioactive decay.

For example, consider the spread of a new viral infection. If each infected person passes it on to a fixed percentage of others, we can use exponential growth models to predict how quickly the outbreak might expand over time, similar to counting how many friends you share an interesting video with and how it spreads.

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