AP Precalculus Review: Unit 1, Part 3

Topics Covered

• Topic 112: Transformations of Functions
• Topic 113: Modeling Situations
• Topic 114: Regression Analysis

Topic 112: Transformations of Functions

General Form and Components

• Function: ( g(x) = a \cdot f(bx + c) + d )
• Additive Transformations (Translations):
  • Horizontal Translation (Inside): Affects x-values
    • If ( c > 0 ): Graph shifts left by c units
    • If ( c < 0 ): Graph shifts right by c units
  • Vertical Translation (Outside): Affects y-values
    • If ( d > 0 ): Graph shifts up by d units
    • If ( d < 0 ): Graph shifts down by d units

Multiplicative Transformations (Dilations)

• Vertical Dilation (Outside):
  • If ( a > 1 ): Vertical stretch
  • If ( 0 < a < 1 ): Vertical compression
• Horizontal Dilation (Inside):
  • Factor of ( \frac{1}{b} )
  • If ( b > 1 ): Horizontal compression
  • If ( 0 < b < 1 ): Horizontal stretch

Reflections

• In the x-axis: ( -f(x) ) flips vertically
• In the y-axis: ( f(-x) ) flips horizontally

Example Problems

• Describing transformations and graphing based on given equations
• Practicing reflection and dilation transformations

Domain and Range Adjustments

• Adjusting domain/range based on transformations

Topic 113: Modeling Situations

Example 1: Box Volume Calculation

• Open box from cardboard by cutting squares
• Equation for volume ( V(x) = x(30 - 2x)(18 - 2x) )
• Domain: ( 0 < x < 9 )

Example 2: Ice Skating Rink Cost

• Initial 2 hours at $10, additional hours $3 each
• Piecewise function for cost over time

Topic 114: Regression Analysis

Using the TI-84 Calculator

• Quadratic Regression

  • Steps to input data and compute regression
  • Predicting values using the stored regression model

• Cubic Regression

  • Entering data, and evaluating at specific points

Example 3: Zebra Population Model

• Using quadratic regression to model and predict zebra population
• Evaluate model for future prediction (e.g., year 2024)

Conclusion

• Importance of mastering transformations as they appear in multiple contexts
• Application of modeling and regression in real-world scenarios

This review covers essential transformations, modeling, and regression techniques necessary for mastering AP Precalculus Unit 1.