Understanding Direct Variation

Overview

This lecture covers the concept of direct variation, including its definition, mathematical representation, applications, and problem-solving with tables, equations, and graphs.

Definition and Properties of Direct Variation

• Direct variation occurs when two quantities have a constant ratio.
• The equation for direct variation is ( y = kx ), where ( k ) is the constant of variation or proportionality.
• If ( x ) increases, ( y ) increases proportionally, and if ( x ) decreases, ( y ) decreases proportionally.

Recognizing and Representing Direct Variation

• Identify direct variation using a table: if ( \frac{y}{x} ) is constant across all pairs, a direct variation exists.
• The equation can also be written as “( y ) is directly proportional to ( x )” or ( y \propto x ).

Application Examples

• Example: A cyclist travels 10 km per hour, with the table showing distance increasing proportionally with time (( d = 10t )).
• Graphing direct variation results in a straight line through the origin.
• Word problems can involve paychecks (( p = kh )), weights, or any scenario where increase/decrease is proportional.

Translating Statements to Equations

• “Failure varies directly as distance” → ( f = kd )
• “Weight varies directly as mass” → ( w = km )
• “Area varies as height” → ( a = kh )

Solving for Constants and Equations

• Substitute given values into ( y = kx ) to solve for ( k ).
• Example: If ( y = 24 ) when ( x = 6 ), then ( k = 4 ) and the equation is ( y = 4x ).

Solving Direct Variation Word Problems

• For paycheck problems: find ( k ) using given hours and pay, then calculate for other hours.
• For “weight on the moon” type questions: use earth weight and moon weight to establish ( k ), then use it to find other values.

Sample Questions and Answers Review

• Recognize the form of direct variation: ( a = kb ).
• Find ( k ) and write the equation given sample values (e.g., if ( y = 28 ), ( x = 2 ), then ( k = 14 ), ( y = 14x )).
• Identify examples and implications (e.g., doubling one variable doubles the other in ( t = 4h )).

Key Terms & Definitions

• Direct variation — Relationship where ( y = kx ) and ( k ) is constant.
• Constant of variation (( k )) — The unchanging ratio ( y/x ) in direct variation.
• Proportionality — When one variable increases/decreases, so does the other, at a constant rate.

Action Items / Next Steps

• Practice five direct variation word problems provided at the end of the lecture.
• Review how to translate word statements into equations.
• Prepare paper and pen for next class exercises.