Overview
This lecture explains the core concepts of direct and inverse variation between two variables, including their definitions, forms, examples, and how to recognize them algebraically.
Direct Variation
- Direct variation occurs when one variable is equal to a constant multiplied by another variable (y = kx).
- Examples include y = x, y = 2x, y = ½x, y = −2x, y = πx.
- If you scale x by a factor, y scales by the same factor (e.g., double x, double y).
- Direct variation can be written in different algebraic forms (e.g., y/x = k, x = (1/k)y).
- If y varies directly with x, then x also varies directly with y (but with a different constant).
Inverse Variation
- Inverse variation occurs when one variable equals a constant divided by another variable (y = k/x).
- Examples include y = 1/x, y = 2/x, y = (1/3)/x, y = −2/x.
- If you multiply x by a factor, y is divided by the same factor (e.g., double x, y halves).
- Inverse variation can also be written as xy = k or x = k/y.
- If y varies inversely with x, then x also varies inversely with y.
Recognizing Variation
- To determine the type of variation, observe how scaling one variable affects the other.
- In direct variation, both variables change by the same factor.
- In inverse variation, one variable changes by a factor and the other changes by its reciprocal.
- Algebraically manipulate equations to standard forms (y = kx for direct, y = k/x for inverse) to identify the variation type.
Key Terms & Definitions
- Direct Variation — A relationship where y = kx, and both variables change by the same factor.
- Inverse Variation — A relationship where y = k/x, and as one variable increases, the other decreases proportionally.
- Constant (k) — The fixed nonzero value relating the variables in direct or inverse variation.
Action Items / Next Steps
- Practice identifying direct and inverse variation in given equations.
- Algebraically manipulate equations to see if they match direct or inverse variation forms.
- Complete any assigned homework on variation problems.