Campaign Ad Analysis and Identifying Linear vs. Non-Linear Functions

Political Campaign Analysis: Trump Campaign Ad

  • Ad Title / Theme: "ARE YOU BETTER OFF THAN YOU WERE 4 YEARS AGO?"
  • Central Messages Regarding the State of America:
    • Global and Economic Status: America is presented as getting worse economically and globally.
    • National Insecurity: The country is framed as being in an insecure state.
    • Military Defense: Safety and national defense are presented as dependent on military power and having weapons of war to protect the nation.
  • Communication Strategies:
    • Uses retrospective economic and security framing by asking viewers to compare present conditions to four years earlier.
    • Emphasizes military strength and defense hardware as central to national stability.

Identifying Linear Functions from Data Tables

  • Core Mathematical Principle:

    • Successive Differences: If equal intervals of input values (e.g., successive weeks) correspond to equal differences in output values, the relation is a linear function.

  • Case Study: Makayla's Hardware Store Earnings:

    • Context: Makayla works part-time at a local hardware store, and her time at work steadily increases over her first five weeks.
    • Data Table:
    • Week 1: 85 dollars85\, \text{dollars}
    • Week 2: 119 dollars119\, \text{dollars}
    • Week 3: 153 dollars153\, \text{dollars}
    • Week 4: 187 dollars187\, \text{dollars}
    • Week 5: 221 dollars221\, \text{dollars}

  • Successive Difference Calculations:

    • Difference between Week 2 and Week 1: 119−85=34119 - 85 = 34
    • Difference between Week 3 and Week 2: 153−119=34153 - 119 = 34
    • Difference between Week 4 and Week 3: 187−153=34187 - 153 = 34
    • Difference between Week 5 and Week 4: 221−187=34221 - 187 = 34

  • Analytical Conclusion:

    • Because all successive differences are equal to 3434, the relationship between the number of weeks and Makayla's earnings is modeled by a linear function with a constant rate of change of 34 dollars34\, \text{dollars} per week.

Identifying Non-Linear Functions

  • Analysis of Radical Equations:
    • Function: f(x)=2x+3f(x) = \sqrt{2x + 3}
    • Linearity Determination: Non-linear function.
    • Mathematical Justification:
    • A linear function must be writeable in slope-intercept form f(x)=mx+bf(x) = mx + b, where the variable xx is raised to the first power (x1x^1) and is not enclosed under a square root or radical sign.
    • In f(x)=2x+3f(x) = \sqrt{2x + 3}, the input variable xx is inside a square root, which represents a fractional exponent of 12\frac{1}{2}, i.e., (2x+3)12(2x + 3)^{\frac{1}{2}}. This creates a variable rate of change, making the function non-linear.