Overview
This lecture covers the concept of direct variation, its formula, how to find the constant of variation, and solving for missing variables using examples.
Direct Variation Basics
- Direct variation describes a relationship where y = kx, with k as the constant of variation.
- "Y varies directly as x" means y changes in proportion to x; if x increases, y increases, and vice versa.
- In the equation, y and x are variables and k is always a fixed number.
- Tables of values or graphs showing both variables increasing indicate direct variation.
- The graph of a direct variation is a straight line passing through the origin (0,0).
Formula Manipulation
- The general formula for direct variation is y = kx.
- To find k, rearrange the formula: k = y / x.
Example Problem 1
- Given: y = 12 when x = 4.
- To find k: k = 12 / 4 = 3.
- Equation of variation: y = 3x.
- Find x when y = 36: 36 = 3x → x = 36 / 3 = 12.
Example Problem 2 (Table of Values)
- For x = 3, 5, 7 and y = 6, 10, 14:
- Calculate k for each pair: k = y / x = 2.
- General equation: y = 2x.
- Find y when x = 9: y = 2 × 9 = 18.
Key Terms & Definitions
- Direct Variation — A relationship where one variable is a constant multiple of another (y = kx).
- Constant of Variation (k) — The fixed number that relates the two variables in direct variation.
- Origin — The point (0,0) where the graph of a direct variation passes through.
Action Items / Next Steps
- Practice finding the constant of variation and writing equations from problem sets.
- Try solving for missing variables in direct variation problems.
- Review the concept of graphing direct variation equations.