When we dive into math in Year 9, we come across different kinds of functions, like linear, quadratic, and exponential functions. Each type helps us understand various situations in the world around us. Today, we’ll focus on exponential functions and learn why they are important for calculating things like growth and decay.

What Are Exponential Functions?

Exponential functions are math expressions that look like this:

$y = a \cdot b^x$

In this equation:

• $a$ is the starting amount,
• $b$ is a positive number we call the base, and
• $x$ is the exponent, usually representing time.

The key thing about exponential functions is that they change at a rate that matches their current value. This makes them great for showing situations where things grow or shrink quickly.

Exponential Growth

Exponential growth is when something increases based on its current size. A common example is population growth.

Imagine a small town that has 100 people to start. If the population grows by 5% each year, we can represent this growth with the formula:

$P(t) = 100 \cdot (1 + 0.05)^t$

Here, $P(t)$ tells us the population after $t$ years.

Let’s see how this looks:

• After 1 year: $P(1) = 100 \cdot (1.05)^1 = 105$
• After 2 years: $P(2) = 100 \cdot (1.05)^2 \approx 110.25$
• After 3 years: $P(3) = 100 \cdot (1.05)^3 \approx 115.76$

As you can see, the population keeps growing more and more each year. This shows how exponential growth works.

Exponential Decay

Now let’s talk about exponential decay. This is when something decreases based on its current size. A good example is radioactive decay.

Imagine we have a substance that loses half of its amount every 5 years. We can show this decay with the formula:

$A(t) = A_0 \cdot (0.5)^{t/5}$

In this formula:

• $A(t)$ is how much is left after $t$ years,
• $A_0$ is the starting amount of the substance.

Let’s say we start with 80 grams of the substance:

• After 5 years: $A(5) = 80 \cdot (0.5)^{5/5} = 80 \cdot 0.5 = 40 \text{ grams}$
• After 10 years: $A(10) = 80 \cdot (0.5)^{10/5} = 80 \cdot 0.25 = 20 \text{ grams}$
• After 15 years: $A(15) = 80 \cdot (0.5)^{15/5} = 80 \cdot 0.125 = 10 \text{ grams}$

Just like with growth, you can see how the decay happens quickly at first. But as time goes on, the amount left decreases more slowly.

Conclusion

In conclusion, exponential functions are very important for showing growth and decay in many areas, like biology and finance. They let us understand rapid changes in things like how populations grow or how substances break down. As you continue learning math, getting a good grasp of exponential functions will help you solve real-world problems easily. So, keep exploring, and enjoy discovering the power of these interesting functions!