How does change over the interval from to ?
decreases by a factor of 7
increases by
decreases by 7
increases by 7

To understand the behavior of over the interval from to , we first evaluate and :

When we compare and , we see that , which means that decreases by a factor of 7 from to .

So, the correct answer is that decreases by a factor of 7.

Now let’s explore a bit further!

Think about how this type of exponential decay occurs in real life. For instance, this pattern appears in various fields, from finance—where compound interest can lead to a reduction in a debt over time when payments are made—to environmental studies, where certain pollutants diminish exponentially in concentration. Understanding this concept can also help us grasp how quickly resources or populations can decrease under certain conditions.

Another fun fact: exponential functions like can often lead to some common mistakes. One of these is assuming that a decrease by a particular factor (like 7 in this case) means the overall rate of change will be linear. Remember, exponential decay is a gradual process! It’s crucial to carefully distinguish between relative changes (like factors) and absolute changes (like differences), as this can lead to misconceptions when interpreting graphs or data in real-world situations.