Direct Variation vs. Inverse Variation - Algebra-2

1. Fundamental Concepts

Definition: Direct Variation: A relationship where one variable is a constant multiple of the other, expressed as , where is a non-zero constant.
Definition: Inverse Variation: A relationship where the product of two variables is a constant, expressed as or , where is a non-zero constant.
Graphical Representation: Direct variation forms a straight line through the origin, while inverse variation forms a hyperbola with asymptotes on the coordinate axes.

Direct Variation Rule:

Inverse Variation Rule:

Application: Used in physics (e.g., Hooke's Law for direct variation), economics (supply and demand for inverse variation)

Problem: Determine if the relationship between and is direct or inverse given .

Validation: Substitute → Original: ; Simplified: ✓

Problem: If varies directly with and when , find the value of when .

Validation: Substitute → Original: ; Simplified: ✓

Identification Strategy: Recognize the form of the equation to determine if it represents direct or inverse variation.
Substitution Method: Use given values to find the constant and then apply it to find unknown values.
Graphical Interpretation: Plot points to visually confirm the type of variation (straight line for direct, hyperbola for inverse).