Section 2-1: Analysis of Growth - Let's Get Growing!

Welcome to Section 2-1, where we'll be exploring the fascinating world of growth analysis! This section focuses on understanding how to measure and interpret change using mathematical tools. Get ready to think critically about data and graphs!

Key Concepts

  • Functions: A function describes how a dependent variable depends on an independent variable. The independent variable is the input, and the dependent variable is the output. For example, the distance you travel depends on the time you're driving; here, time is independent, and distance is dependent.
  • Percentage Change: Also known as relative change, it shows the percentage increase in a function's value from one point to another. The formula is:

$$\text{Percentage Change} = \frac{\text{Change in Function}}{\text{Previous Function Value}} \times 100\%$$

Let's consider an example: Suppose a company's revenue increased from $1 million to $1.2 million in a year. The percentage change is: $$\frac{1.2 - 1}{1} \times 100\% = 20\%$$. That's some impressive growth!

  • Average Growth Rate: This is the change in the function divided by the change in the independent variable over a specific interval.

$$\text{Average Growth Rate} = \frac{\text{Change in Function}}{\text{Change in Independent Variable}}$$

Imagine the population of a town increased from 10,000 to 11,000 over 5 years. The average growth rate is $$ rac{11000 - 10000}{5} = 200 \text{ people per year}$$. Pretty neat, right?

  • Interpolation: Estimating unknown values within a known range of data points. We use the average growth rate to make these estimations.
  • Extrapolation: Estimating unknown values beyond the known range of data points, again using the average growth rate. Be careful with extrapolation, though, as it can lead to inaccurate predictions if trends change!

Putting It All Together

Let's say we have some data about the number of students enrolled in Professor Baker's Math Class over the years:

  • 2020: 50 students
  • 2022: 60 students

We can calculate the percentage change in enrollment from 2020 to 2022:

$$\text{Percentage Change} = \frac{60 - 50}{50} \times 100\% = 20\%$$

And the average growth rate:

$$\text{Average Growth Rate} = \frac{60 - 50}{2022 - 2020} = \frac{10}{2} = 5 \text{ students per year}$$

If we wanted to interpolate the enrollment for 2021, we could estimate it by adding the average growth rate to the 2020 enrollment: 50 + 5 = 55 students.

And if we wanted to extrapolate for 2024, we'd add the average growth rate twice to the 2022 number: 60 + (5*2) = 70 students.

Practice Makes Perfect!

Now it's your turn! Try calculating percentage changes and average growth rates for different scenarios. Remember to think about the independent and dependent variables, and be mindful of the limitations of extrapolation. Good luck, you've got this!