Welcome to Week 2 of MAT137! This week marks an exciting transition as we move from basic algebraic manipulation to mathematical modeling. We are looking at Linear Equations not just as lines on a graph, but as powerful tools to interpret information and predict future values.

Understanding Linear Models (Sections 2.1 & 2.2)

In Sections 2.1 and 2.2, we focus on the fundamental building block of modeling: the linear equation. A linear model describes a relationship between two variables that changes at a constant rate.

The most common form you will use is the Slope-Intercept Form:

$$ f(x) = mx + b $$

Where:

  • $x$ is the Independent Variable (the input, often time or quantity).
  • $f(x)$ or $y$ is the Dependent Variable (the output, such as cost, distance, or profit).
  • $m$ is the Slope (the Rate of Change).
  • $b$ is the y-intercept (the Initial Value when $x=0$).

Key Concepts for the Week

As you review the attached class notes and video lecture, pay close attention to these three core concepts:

  1. Slope as Rate of Change:
    In a pure math problem, slope is "rise over run." In the real world, it represents how fast something is changing. For example, if you are modeling the cost of production, the slope might represent the cost per unit produced. $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
  2. The Initial Value ($y$-intercept):
    This is where the graph crosses the vertical axis. In a business model, the $y$-intercept often represents fixed costs or overhead—expenses you have to pay even if you produce zero units ($x=0$).
  3. Predictions (Interpolation vs. Extrapolation):
    Once we have our equation, we can plug in values for $x$ to predict what will happen in the future.
    • Interpolation is predicting a value inside the range of our data points.
    • Extrapolation is predicting a value outside our current data range (predicting the future).

Example: Modeling Cost

Imagine a service with a flat fee of $50 plus $10 per hour. We can model the total cost ($C$) based on hours worked ($h$) as:

$$ C(h) = 10h + 50 $$

Here, the slope is $10$ (dollars/hour) and the intercept is $50$ (start-up fee).

Weekly Tasks

Please ensure you download the class notes below and watch the lecture video before attempting the quiz. These resources cover the specific step-by-step methods for calculating regression lines and interpreting scatter plots.

Let's have a great week analyzing data! If you get stuck on calculating the slope or interpreting the variables, remember to check the discussion board.