Part 3: Critical Thinking and Analysis 1. Compare the rates of change for two different types of functions (e.g., quadratic vs. exponential) at specific intervals Discuss how the nature of the function affects the interpretation of the rate of change. 2. Create your own real-world scenario involving one of the function types studied Write a word problem, solve it, and explain its relevance.

## Step 1: Defining Rate of Change<br />### Rate of change describes how much one quantity changes in relation to another. For functions, it often represents how much the output ($y$ or $f(x)$) changes for a given change in the input ($x$).<br /><br />## Step 2: Comparing Quadratic and Exponential Rates of Change<br />### Let's compare a quadratic function, $f(x) = x^2$, and an exponential function, $g(x) = 2^x$. Consider the intervals $[0, 1]$ and $[3, 4]$.<br /><br />### For $f(x)$:<br />The average rate of change over $[0, 1]$ is $\frac{f(1) - f(0)}{1 - 0} = \frac{1 - 0}{1} = 1$.<br />The average rate of change over $[3, 4]$ is $\frac{f(4) - f(3)}{4 - 3} = \frac{16 - 9}{1} = 7$.<br /><br />### For $g(x)$:<br />The average rate of change over $[0, 1]$ is $\frac{g(1) - g(0)}{1 - 0} = \frac{2 - 1}{1} = 1$.<br />The average rate of change over $[3, 4]$ is $\frac{g(4) - g(3)}{4 - 3} = \frac{16 - 8}{1} = 8$.<br /><br />### Notice that while both functions have the same rate of change over $[0,1]$, the exponential function's rate of change grows much faster than the quadratic function's as $x$ increases.<br /><br />## Step 3: Impact of Function Nature on Interpretation<br />### The nature of the function significantly impacts the interpretation of the rate of change. Quadratic functions have a constantly increasing rate of change, but this increase is linear. Exponential functions, however, have an exponentially increasing rate of change, leading to much more rapid growth.<br /><br />## Step 4: Real-World Scenario with Exponential Growth<br />### Consider bacterial growth. Suppose a bacteria population doubles every hour. If we start with 100 bacteria, the population after $t$ hours can be modeled by $P(t) = 100 \cdot 2^t$. How many bacteria will there be after 3 hours?<br /><br />### Solution:<br />We need to find $P(3)$.<br />$P(3) = 100 \cdot 2^3 = 100 \cdot 8 = 800$.<br /><br />### Relevance:<br />This example demonstrates the rapid growth characteristic of exponential functions. In real-world scenarios like bacterial growth, this rapid increase can be crucial for understanding phenomena like infections or the effectiveness of treatments.