Section 2-1: Analysis of Growth

Welcome to Professor Baker's Math Class! In this section, we will delve into the fascinating world of analyzing growth. We will explore how to measure growth, represent it visually with graphs, and be aware of misleading representations. Let's get started!

Measurements of Growth

The core question we're addressing is: How fast is it changing? Understanding this involves several key concepts:

  • Functions: A function describes how a dependent variable depends on an independent variable. Think of it this way: if one quantity (or variable) depends on another, the latter is the independent variable, and the former is the dependent variable. The independent variable is the input value, and the dependent variable is the output value of a function.

For example, if you work for an hourly wage, the number of hours you work is the independent variable (input), and your pay is the dependent variable (output). Therefore, your pay is a function of the hours you work.

Percentage Change

The percentage change (or relative change) in a function measures the percentage increase in the function from one value of the independent variable to another. The formula is:

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

Let's consider an example: Suppose the U.S. population was 3.93 million in 1790 and 5.31 million in 1800. The percentage change in population from 1790 to 1800 is calculated as follows:

$$Percentage\ Change = \frac{5.31 - 3.93}{3.93} \times 100\% = \frac{1.38}{3.93} \times 100\% \approx 35\%$$.

Average Growth Rate

The average growth rate of a function over an interval is the change in the function divided by the change in the independent variable. The formula is:

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

For instance, if the population of Russia declined from 146 million in 2000 to 143 million in 2007, the average growth rate is:

$$Average\ Growth\ Rate = \frac{143 - 146}{2007 - 2000} = \frac{-3}{7} \approx -0.429\ million\ per\ year$$.

This means that, on average, the population declined by approximately 429,000 people per year during that period.

Interpolation and Extrapolation

  • Interpolation is estimating unknown values between known data points using the average growth rate.
  • Extrapolation is estimating unknown values beyond known data points using the average growth rate.

Remember that extrapolation should be used cautiously, as it assumes that the trend will continue indefinitely, which may not always be the case!

Keep practicing these concepts, and you'll become a master of analyzing growth. Good luck with your Section 2-1 Assignment!