AP Precalculus Formula Sheet Notes

Unit One: Polynomial and Rational Functions

Key Concepts

1. Average Rate of Change

   • Measures change in function value over an interval relative to the change in x.

2. Increasing/Decreasing Functions

   • Increasing: Positive slope; function values rise as x increases.
   • Decreasing: Negative slope; function values fall as x increases.
   • Constant: No increase or decrease over interval.

3. Concavity

   • Concave Up: Graph appears like a smile; slope is increasing.
   • Concave Down: Graph appears like a frown; slope is decreasing.

4. Types of Functions

   • Linear: Constant rate of change; y = mx+b.
   • Quadratic: Linear pattern of change; f(x) = ax² + bx + c.
   • Polynomial (degree n): nth order rates of change are constant.

5. Local (Relative) Maximum/Minimum

   • Maximum: Function changes from increasing to decreasing.
   • Minimum: Function changes from decreasing to increasing.

6. Point of Inflection

   • Point where concavity changes; graph switches curvature.

7. Complex Roots

   • Complex conjugate of complex root is also a root.

8. Even and Odd Functions

9. Polynomial End Behavior

   • Depends on degree and sign of leading coefficient.

10. Rational Functions

    • Zeroes of R are zeroes of P not also zeroes of Q.

11. Hole in the Graph

    • Occurs when P and Q equal zero at x = c; function is undefined but may be simplified.

12. Vertical Asymptote

    • Occurs where Q(x) = 0 while P(x) is non-zero.

13. End Behavior of Rational Functions

    • Indicates if function approaches 0, infinity, or negative infinity.

14. Binomial Expansion

    • Use box multiplication or Pascal's Triangle.

15. Transformations of Functions

Unit Two: Exponential and Logarithmic Functions

1. Base Logarithmic Function

2. Exponential Rules

3. Logarithmic Rules

   • Assumed base 10 if no subscript.
   • Natural logs (ln) use base e.

4. Function Composition

5. Function Inverses

Unit Three: Trigonometry and Polar Coordinates

1. Standard Position

   • Vertex at origin; ray on positive x-axis.

2. Arc Length Formula

3. Unit Circle (Radian Angles)

4. Sine, Cosine, and Tangent

5. Special Right Triangles

6. Trigonometric Identities

7. Graphs of Trigonometric Functions

   • Equation: y = Asin(B(x + c)) + D.
   • A: Amplitude; Period: 2/B; Phase Shift: C; Midline: y = D.

8. Vertical Asymptotes

9. Inverse Trigonometry

10. Polar Coordinates

    • Important formulas and applications.

11. Modeling

    • Semi-log plots indicate exponential functions.
    • Residual plot patterns indicate model appropriateness.