Study Notes on Functions and Linear Sequences

Functions

  • Definition: Functions map input values to output values.

  • Key Concept: Determine output by applying a specified rule to input.

Linear Sequence

  • Definition: A sequence that increases by a constant amount (difference).

  • Example of linear sequence: 5,7,9,115, 7, 9, 11 (difference = +2+2).

Finding Output Differences

  • Important Step: Calculate the difference between outputs to identify patterns.

  • Example Calculation: For input = −1,0,1,2{-1, 0, 1, 2}, outputs are −2,1,4,7{-2, 1, 4, 7}; differences: +3,+3,+3+3, +3, +3.

Equation for Rule

  • General Form: y=(x×d)+cy = (x \times d) + c

    • xx: Input value

    • yy: Output value

    • dd: Constant difference (slope)

    • cc: Constant added value (initial value or y-intercept).

Example Calculation

  • For y=−2y = -2 given x=−1x = -1: −2=(−1×3)+1-2 = (-1 \times 3) + 1

  • For y=4y = 4 given x=1x = 1: 4=(1×3)+14 = (1 \times 3) + 1

Mathematical Notation

  • Coordinates: General form (x,y)(x, y)

  • Example: Origin is (0,0)(0, 0)

  • Emphasis on constant differences and sequences to derive function rules.