Types of Variations

Variations Overview

• Week 1 Day 3 (Second Quarter)

• Introduces types of variations: Direct Variation, Inverse Variation, Joint Variation, and Combined Variation.

Direct Variation

Definition

• If y and x increase or decrease by the same factor, then:

  • Mathematical representation:( y = kx ) where ( k ) is the constant of variation (constant of proportionality).

Interpretations

• Statements can be expressed as:

  • "y varies directly as x"

  • "y is directly proportional to x"

  • "y is proportional to x"

Formula 1

• Given: ( y_1 x_1 = y_2 x_2 )

• Solution:

  • Example: If ( y ) varies directly as ( x ), and ( y = 24) when ( x = 3 ), find ( x ) when ( y = 32 ).

  • Steps:

    • ( y_1 = 24 )

    • ( x_1 = 3 )

    • ( y_2 = 32 )

    • ( \frac{y_1}{x_1} = \frac{y_2}{x_2} ) implies ( \frac{24}{3} = \frac{32}{x_2} )

    • Result: ( x_2 = 4 )

Formula 2

• Reiterates ( y = kx ) where ( k ) is constant.

• Example:

  • Same scenario, deriving k:

    • ( k = \frac{y}{x} \rightarrow k = \frac{24}{3} = 8 )

    • Solve ( 32 = 8x ), gives ( x = 4 )

Graphical Representation

Example

• Distance as a function of time (direct variation):

  • Data:

    • Time (hr): 1, 2, 3, 4, 5

    • Distance (km): 10, 20, 30, 40, 50

  • Show constant ratio: ( \frac{d}{t} = 10 ).

  • Graph plotted demonstrates a linear relationship.

Equation Derivation

• From data show that:

  • ( d = 10t )

  • For 8 hours, ( d = 10(8) = 80 ) km.

  • For 10.5 hours, ( d = 10(10.5) = 105 ) km.

Inverse Variation

Definition

• It occurs when a pair of numbers maintains a constant product.

• Changes inversely, means:

  • If ( x ) increases, ( y ) decreases.

• Expressed as ( y = \frac{k}{x} ).

Statements

• Can be phrased as:

  • "y varies inversely as x"

  • "y is inversely proportional to x"

Formula

• ( x_1 y_1 = x_2 y_2 )

  • Example:

    • If ( y = 12 ) when ( x = 2 ). Find ( y ) when ( x = 8 ).
      - Solution shows ( 2\cdot12 = 8\cdot y_2 \rightarrow y_2 = 3 )

Joint Variation

Definition

• A relationship where one variable varies directly to the product of two or more other variables.

  • Expressed as ( z = kxy ).

Example

• Given ( z = 144 ), ( x = 16 ), ( y = 18 ).

• Find ( y ) when ( z = 24 ), ( x = 16 ).

Combined Variation

Definition

• Describes variables depending on others, varies directly with some, inversely with others.

  • Statement: ( z = k \frac{x}{y} ).

Examples and Problem-Solving

• Finding values for various scenarios involves calculation of constants, maintaining a direct/inverse relationship for varied values.