AP Precalculus Formula Sheet

Unit One: Polynomial and Rational Functions

Average Rate of Change

  • Fundamental formula necessary for analysis, calculated as ( \frac{f(b) - f(a)}{b - a} ) for two points ( (a, f(a)) ) and ( (b, f(b)) ). This measures how much the function value changes over an interval relative to the change in x.

Increasing/Decreasing Functions:

  • Increasing: The function values rise as ( x ) increases, indicating a positive slope.

  • Decreasing: The function values fall as ( x ) increases, indicating a negative slope.

  • A function is constant if it neither increases nor decreases over an interval.

Concavity:

  • Concave Up: The graph of the function is shaped like a cup, indicating that the slope of the tangent is increasing (local minimum).

  • Concave Down: The graph of the function is shaped like an upside-down cup, indicating that the slope of the tangent is decreasing (local maximum).

Linear vs. Quadratic Functions:

  • Linear functions maintain a constant rate of change, represented as ( f(x) = mx + b ) where ( m ) is the slope.

  • Quadratic functions, represented as ( f(x) = ax^2 + bx + c ), exhibit a changing rate of change, where the first derivative is linear. A degree ( n ) polynomial has constant nth order rates of change with evenly spaced x-values and varies in behavior based on degree and leading coefficient.

Local and Relative Maxima/Minima:

  • Maximum: This occurs at points where the function transitions from increasing to decreasing, indicating a peak.

  • Minimum: This occurs at points where the function transitions from decreasing to increasing, indicating a trough.

  • Critical points are found by setting the first derivative to zero and analyzing sign changes.

Point of Inflection:

  • Points of inflection are locations on the graph where the concavity changes, which occurs where the second derivative is zero or undefined.

Complex Roots:

  • If a complex number ( z = a + bi ) is a root of a polynomial function, its complex conjugate ( z = a - bi ) must also be a root due to the coefficients being real numbers. This indicates that polynomial functions with real coefficients have complex roots that appear in conjugate pairs.

Even and Odd Functions:

  • Even Function: Functions that satisfy ( f(-x) = f(x) ) show symmetry across the y-axis, like ( f(x) = x^2 ).

  • Odd Function: Functions that satisfy ( f(-x) = -f(x) ) exhibit symmetry about the origin, such as ( f(x) = x^3 ).

End Behavior of Polynomials:

  • The end behavior of polynomial functions depends on the degree (whether even or odd) and the sign of the leading coefficient. For example, even degree polynomials with a positive leading coefficient rise to infinity in both directions, while odd degree polynomials with a negative leading coefficient fall to negative infinity as they approach negative infinity for x.

Rational Functions

  • Rational Function Definition: A rational function is expressed in the form ( R(x) = \frac{P(x)}{Q(x)} ), where both P and Q are polynomials.

  • Domain of R: Determined by all x-values for which ( Q(x)
    eq 0 ) to avoid undefined outputs.

Vertical Asymptotes:

  • Vertical asymptotes occur at x-values where ( Q(x) = 0 ) while ( P(x) ) is non-zero, showing where the function approaches infinity.

  • Holes: A hole exists in the graph if both P and Q equal zero at x = C, indicating that the function is undefined, but can often be simplified by canceling common factors.

Zeros of a Rational Function:

  • Zeros are values at which the numerator is zero but the denominator is non-zero, indicating x-intercepts on the graph.

End Behavior:

  • End behavior can be analyzed by comparing the degrees of the numerator and denominator polynomials, indicating whether the rational function approaches 0, infinity, or negative infinity as ( x ) approaches infinity.

Binomial Expansion:

  • The process of expanding expressions of the form ( (x + y)^n ) can be effectively understood through Pascal's Triangle, allowing for the derivation of coefficients rather than relying on memorization.

Unit Two: Exponential and Logarithmic Functions

Logarithm Definition:

  • A logarithm indicates the exponent to which a base must be raised to yield a given number, expressed as ( \log_b(A) = X ) which implies ( b^X = A ).

Fundamental Logarithmic Identities:

  • Key properties include ( b^{\log_b(X)} = X ) and ( \log_b(b^X) = X ), which form the foundation for manipulating expressions involving logarithms.

Properties of Exponential Functions:

  • Important properties of exponential functions include the product, quotient, and power rules, which govern how to simplify complex expressions involving exponents.

Function Composition and Inverses:

  • If G is the inverse of function F, then it holds true that ( F(G(x)) = x ) and ( G(F(x)) = x ). This relationship is crucial for understanding the behavior of functions and their reversibility.

Trigonometry and Polar Coordinates

Basic Trigonometric Functions:

  • Sine (sin) represents the vertical displacement, while Cosine (cos) represents the horizontal displacement in a right triangle based on angle measurement.

Key Triangles:

  • Familiarity with special triangles such as the 30-60-90 triangle and 45-45-90 triangle is crucial for solving problems involving trigonometric ratios quickly.

Arc Length Formula:

  • The arc length of a sector can be calculated with the formula: Arc length = radius * angle in radians (( R * \theta )).

Trigonometric Identities:

  • Mastery includes even-odd identities, reciprocal identities, the Pythagorean identity, angle addition formulas, and double angle formulas. Knowledge of generating identities through division and substitutions is also essential for solving complex trigonometric equations.

Graphing Trigonometric Functions:

  • Understanding graph transformations includes concepts like amplitude (|A|), period (2π/B), phase shift (C), and midline (y = D) is essential for sketching accurate graphs of trigonometric functions.

Vertical Asymptotes:

  • Vertical asymptotes are identified in tangent and secant functions at x-values where cosine equals zero (e.g., ( \frac{\pi}{2}, \frac{3\pi}{2} )). Their significance is crucial in determining the behavior of these functions.

Polar Coordinates and Modeling

Polar Coordinates:

  • The relationship in polar coordinates is defined by ( x = r * cos(θ) ) and ( y = r * sin(θ) ), with ( r^2 = x^2 + y^2 ). Understanding this transformation between Cartesian and polar coordinates is essential.

  • The radius can be negative, leading to unique points on the polar plot corresponding to angles in different quadrants.

Modeling Skills:

  • Students should possess the ability to conduct regressions and analyze graphs effectively, with an understanding that semi-log plots indicate exponential behavior.

Final Notes:

  • Students are encouraged to review all major concepts thoroughly and make use of detailed lesson playlists for clarification and deeper understanding of the material.