Topic 1: Linear Functions - Rate of Change

Connections Between Arithmetic Sequences and Linear Functions

  • Definitions and Fundamental Concepts:

    • An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant value is known as the common difference (dd).
    • A linear function is a polynomial function of degree 11 or 00, represented graphically as a straight line.
    • The discrete terms of an arithmetic sequence map directly to specific points on a continuous linear function.

  • Comparison of Key Components:

    • Common Difference vs. Slope: The common difference (dd) of an arithmetic sequence represents the exact same constant rate of growth or decay as the slope (mm) of a linear function.
    • Domain Restrictions:
    • Arithmetic sequences have discrete domains limited to positive integers or whole numbers (n∈{1,2,3,… }n \in \{1, 2, 3, \dots\}).
    • Linear functions have continuous domains extending across real numbers (x∈Rx \in \mathbb{R}).

  • Formulas and Algebraic Equivalences:

    • Arithmetic Sequence Explicit Formula:     an=a1+(n−1)da_n = a_1 + (n - 1)d
    • Linear Function Slope-Intercept Form:     f(x)=mx+bf(x) = mx + b
    • Expanding the explicit arithmetic sequence formula reveals its linear form:     an=d⋅n+(a1−d)a_n = d \cdot n + (a_1 - d)     where the slope m=dm = d and the y-intercept b=a1−db = a_1 - d

Rate of Change and Slope of Linear Functions

  • Definition of Rate of Change (Slope):

    • The rate of change, or slope (mm), quantifies the steepness and direction of a line on a coordinate plane.
    • It is mathematically defined as the ratio of vertical change (Δy\Delta y, or rise) to horizontal change (Δx\Delta x, or run).

  • The Slope Formula:

    • For any two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on a line, the slope formula is:     m=ΔyΔx=y2−y1x2−x1m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

  • Classifications of Slope:

    • Positive Slope (m>0m > 0): The line ascends from left to right (Δy\Delta y and Δx\Delta x share the same sign).
    • Negative Slope (m<0m < 0): The line descends from left to right (Δy\Delta y and Δx\Delta x have opposite signs).
    • Zero Slope (m=0m = 0): The line is horizontal. There is no vertical change (Δy=0\Delta y = 0), while horizontal change is non-zero (Δx≠0\Delta x \neq 0).
    • Undefined Slope (m=undefinedm = \text{undefined}): The line is vertical. There is vertical change (Δy≠0\Delta y \neq 0), but no horizontal change (Δx=0\Delta x = 0), resulting in division by zero.

Practice Problems: Determining Rate of Change from Graphs

Practice problems for determining the rate of change (slope) from graphs

  • Problem 1 Solution:

    • Given Coordinates: (3,10)(3, 10) and (9,10)(9, 10)
    • Identification of Line: Horizontal line passing through y=10y = 10
    • Step-by-Step Calculation:
    • Calculate change in yy: Δy=y2−y1=10−10=0\Delta y = y_2 - y_1 = 10 - 10 = 0
    • Calculate change in xx: Δx=x2−x1=9−3=6\Delta x = x_2 - x_1 = 9 - 3 = 6
    • Apply slope formula:       m=06=0m = \frac{0}{6} = 0
    • Conclusion: The slope (rate of change) is m=0m = 0

  • Problem 2 Solution:

    • Given Coordinates: (3,5)(3, 5) and (9,12)(9, 12)
    • Step-by-Step Calculation:
    • Calculate rise (Δy\Delta y): Δy=12−5=7\Delta y = 12 - 5 = 7
    • Calculate run (Δx\Delta x): Δx=9−3=6\Delta x = 9 - 3 = 6
    • Apply slope formula:       m=y2−y1x2−x1=12−59−3=76m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 5}{9 - 3} = \frac{7}{6}
    • Conclusion: The slope (rate of change) is m=76m = \frac{7}{6}

  • Problem 3 Solution:

    • Given Coordinates: (1,7)(1, 7) and (13,2)(13, 2)
    • Step-by-Step Calculation:
    • Calculate rise (Δy\Delta y): Δy=2−7=−5\Delta y = 2 - 7 = -5
    • Calculate run (Δx\Delta x): Δx=13−1=12\Delta x = 13 - 1 = 12
    • Apply slope formula:       m=y2−y1x2−x1=2−713−1=−512m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 7}{13 - 1} = -\frac{5}{12}
    • Analysis of Potential Pitfalls:
    • Inverting rise and run yields −125-\frac{12}{5}, which is incorrect. The numerator must always represent the vertical change (Δy=−5\Delta y = -5), and the denominator must represent the horizontal change (Δx=12\Delta x = 12).
    • Conclusion: The correct slope (rate of change) is m=−512m = -\frac{5}{12}

  • Problem 4 Solution:

    • Given Coordinates: (2,−9)(2, -9) and (10,−4)(10, -4)
    • Step-by-Step Calculation:
    • Calculate rise (Δy\Delta y): Δy=−4−(−9)=5\Delta y = -4 - (-9) = 5
    • Calculate run (Δx\Delta x): Δx=10−2=8\Delta x = 10 - 2 = 8
    • Apply slope formula:       m=y2−y1x2−x1=−4−(−9)10−2=58m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-9)}{10 - 2} = \frac{5}{8}
    • Analysis of Potential Pitfalls:
    • Inverting the slope formula gives 85\frac{8}{5}, which is incorrect. Rise (Δy=5\Delta y = 5) must be placed in the numerator, and run (Δx=8\Delta x = 8) in the denominator.
    • Conclusion: The correct slope (rate of change) is m=58m = \frac{5}{8}