Module 1 Notes: Linear Functions and Basic Modeling

Module 1: Linear Function — Comprehensive Notes

• Context and Purpose: Linear functions model lines to approximate real-world data in business, for prediction, cost/revenue/profit analysis, and basic market models (supply/demand).

• Basic Definitions: A line is defined by two points, or one point and a slope. The independent variable (input) is typically x, and the dependent variable (output) is y. Points are ordered pairs $(x, y)$.

• Slope and Interpretation: Slope $m$ (rise over run) quantifies how y changes per unit change in x. Formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1} = \frac{\Delta y}{\Delta x}$. Positive slope means y increases with x; negative means y decreases; zero slope is horizontal; undefined slope is vertical.

• Intercepts and Lines: The slope-intercept form is $y = m x + b$, where $b$ is the y-intercept (value of y when $x = 0$). The point-slope form is $y - y<em>1 = m(x - x</em>1)$. The x-intercept is where $y = 0$, so $x = -\frac{b}{m}$ (if $m \neq 0$). Horizontal lines are $y = c$ (slope $0$); vertical lines are $x = c$ (slope undefined).

• Domain Concepts: The domain (D) is the set of allowable x-values. In business, quantities are often nonnegative, restricting the domain (e.g., $[0, \infty)$).

• Real-World Modeling: Linear models simplify complex relationships for cost, revenue, profit, and supply/demand. They help solve for total cost, breakeven points (where revenue equals cost, $\pi = 0$), and equilibrium (supply-demand intersection).

• Equilibrium: The point where supply and demand lines intersect, yielding equilibrium quantity $(q<em>e)$ and price $(p</em>e)$. Fixed costs are one-time; variable costs are per-unit. Profit is $\pi = R - C$. Breakeven quantity: $q = \frac{FC}{p - VC}$.

• Estimating from Data: Best-fit lines (least-squares regression) minimize error when fitting a line to data points. Excel can calculate the slope $(m)$, intercept $(b)$, and correlation coefficient $( \rho )$. The correlation coefficient $( \rho \in [-1, 1] )$ indicates the strength of the linear relationship ($\pm 1$ is strong, $0$ is weak). Desmos is a tool for visualization.

• Model Interpretation and Cautions: Linear models are approximations; their validity may be limited to specific domains. Always discuss domain, error, and bias. Recognize that real-world relationships may require nonlinear or piecewise models for better accuracy.

• Key Equations Summary:

  • Slope: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

  • Slope-intercept form: $y = m x + b$

  • Point-slope form: $y - y<em>1 = m(x - x</em>1)$

  • X-intercept: $x = -\frac{b}{m}$

  • Profit: $\pi = R - C$

  • Breakeven quantity: $q = \frac{FC}{p - VC}$

  • Correlation coefficient: $\rho = \frac{\text{Cov}(X,Y)}{\sigma<em>X \sigma</em>Y}$

• Tools and Skills Emphasized: Excel for calculations, Desmos for visualization and exploration of equations and inequalities.

• Assignments: Course assignments include a multi-paragraph discussion post about personal math history and career applications, emphasizing professional language and the use of tools like Desmos.