Overview
This lecture covers how to construct and interpret a linear cost function from data, including finding and interpreting the slope and intercept, and strategies for graphing.
Writing Linear Cost Equations
- Linear cost functions relate production quantity (x) to total cost (C).
- Given: At 50 rackets, cost is $3,855; at 60 rackets, cost is $4,245.
- Treat cost as y and production as x for calculation.
- Coordinates: (50, 3855) and (60, 4245).
Finding the Equation
- Slope (m) formula: (y₂ - y₁) / (x₂ - x₁) = (4245 - 3855) / (60 - 50) = 390 / 10 = 39.
- Use point-slope form: y - y₁ = m(x - x₁).
- Applying (50, 3855): y - 3855 = 39(x - 50).
- Rearranged: y = 39x + 1905.
- The cost function: C(x) = 39x + 1905.
Interpreting Slope and Intercept
- Slope (39): Cost increases by $39 for each extra racket produced (variable cost).
- Y-intercept (1905): Fixed cost when zero rackets are produced.
Graphing the Cost Function
- Use original data points and the y-intercept for accurate graphing.
- Domain: x ≥ 0, since negative production is not meaningful.
- Do not extend the line to negative values of x.
Key Terms & Definitions
- Linear Function — A function with a constant rate of change, graphing to a straight line.
- Slope (m) — The change in cost per unit increase in production.
- Y-intercept (b) — The fixed cost when production is zero.
- Fixed Cost — Costs incurred even if nothing is produced.
- Variable Cost — Costs that change with the number of units produced.
Action Items / Next Steps
- Practice writing linear cost equations from word problems.
- Graph cost functions using the original data points and y-intercept.
- Interpret slope and y-intercept in context for future problems.