GRE Prep Lecture: Function Shapes

Introduction

• Instructor: Tyler from Manhattan Prep
• Topic: Function Shapes
• Format: Live Zoom session to be uploaded on YouTube later
• Participants from various locations (NY, TX, CA)

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Overview of Function Evaluation

• Using visualization for function evaluation instead of just picking values for x
  • Fundamental shapes include:
    • Linear Shape: y = x (straight line)
    • Quadratic Shape: y = x² (u-shaped parabola)
    • Cubic Shape: y = x³ (cubic curve)

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Function Comparison Technique

• Example Problem: x³ < x
  • Visualization helps determine where one function is smaller than another
  • Identify the intersection points: x = -1 and x = 1
  • Resulting domains: positive fractions and negative values less than -1
  • Result: Answer is D (sometimes A is bigger, sometimes B is bigger)

Steps to Visualize

1. Identify the shape for each function.
2. Determine where one curve lies beneath the other.
3. Check values in the determined domains for comparisons.

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Further Examples

• Next Problem: n⁵ < n³
  • Simplifying gives similar inequalities to the previous example
  • Results in similar domains, confirming Answer as D

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Odd vs. Even Powers

• Comparison between Odd Powers: x vs x³

  • Linear (straight line) vs cubic shape: Identify intersection points and domain intervals
  • Positive inputs give different outcomes depending on the intervals chosen

• Comparison between Even Powers: x² vs x⁴

  • Identify points of equality: 1, 0, -1
  • Square is lesser in specific domains depending on positive values

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Modifiers Impacting Functions

• Absolute Value:

  • Wrap functions to ensure they cannot drop below the x-axis
  • Example: |x³| creates a parabolic effect

• Negatives:

  • Any function multiplied by -1 flips it vertically across the x-axis

Practice Problems

1. |x³| < 64

• Result: Between -4 and 4
• Answer: D

2. Alt Function Comparison with Absolute Values

• Similar process holds true for mixtures of functions that include absolute values.

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Recap & Closing Thoughts

• Familiarize with the basic shapes: linear, parabolic, cubic
• Understand modifiers and their impacts on the function shapes
• Use visualizations to compare functions effectively

Additional Resources

• Link for additional GRE prep hours
• Access to introductory courses

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Final Notes

• Tyler encourages continuous study and participation in GRE preparations.
• Wishing everyone the best in their studies!