The question on the screen asks which function represents exponential decay, and the image shows four different functions plotted on Desmos. Let's break down each function to determine which one represents exponential decay.

Identifying Exponential Decay:

An exponential decay function is typically of the form:

$f(x) = a \cdot b^x \quad \text{where} \quad 0 < b < 1$

Now, let's evaluate each function in the image:

1. $f(x) = 0.25(0.6)^x$:

   • This function has a base of 0.6, which is less than 1. This indicates exponential decay.

2. $f(x) = 412 + 1.03^x$:

   • This function has a base of 1.03, which is greater than 1. This indicates exponential growth.

3. $f(x) = 268(0.86)^x$:

   • This function has a base of 0.86, which is less than 1. This also indicates exponential decay.

4. $f(x) = 18 + 0.9x$:

   • This is a linear function because it is of the form $f(x) = mx + b$, so it is neither exponential decay nor growth.

Conclusion:

Both $f(x) = 0.25(0.6)^x$ and $f(x) = 268(0.86)^x$ represent exponential decay. However, the first option, $f(x) = 0.25(0.6)^x$, has a faster rate of decay due to the smaller base (0.6).

It seems the correct answer is $f(x) = 0.25(0.6)^x$, which corresponds to the first function.

Would you like any further explanation or details?

Here are 5 related questions to expand on this concept:

1. How does the base value in exponential functions affect the rate of decay or growth?
2. What are the characteristics of exponential growth functions compared to decay functions?
3. Can exponential decay functions ever reach zero? Why or why not?
4. How do you graph an exponential function manually without software?
5. What are real-world applications of exponential decay?

Tip: Exponential decay models are often used in real-life scenarios such as radioactive decay, depreciation of assets, and population decline. Understanding the base of the exponential function is key to interpreting these models.