AP Precalculus Exam Cram

AP Precalculus Study Guide

This document serves as a comprehensive reference for critical precalculus concepts, including definitions, theorems, and properties essential for the AP Exam.

Definitions, Theorems, and Properties

• Average Rate of Change (AROC): The average rate at which a function changes between two points.
     - Formula: If we have two points (x1, y1) and (x2, y2), the AROC is calculated as:
     $ext{AROC} = rac{y_2 - y_1}{x_2 - x_1}$
     - Types of Rates of Change:
       - Positive rate of change: The function is increasing between the two points.
       - Negative rate of change: The function is decreasing between the two points.
       - Increasing rate of change: The slope of the function is getting steeper.
       - Decreasing rate of change: The slope of the function is becoming less steep.
       - Change from increasing to decreasing or vice versa: Indicates a local maximum or minimum.
  

• Even and Odd Functions:
     - Odd Function: A function is odd if it passes through the origin.
     - Even Function: A function is even if it reflects over the y-axis.

Curvature and Behavior

• Curvature and Rate of Change:
     - If a function decreases with increasing rate of change, it is concave down.
     - If it increases with increasing rate of change, it is concave up.
     - Decrease with decreasing rate of change indicates concavity down.
     - Increase with decreasing rate of change indicates concavity up.

Intersection and Asymptotic Behavior

• Intersections:
     - A function crosses the x-axis linearly.
     - It bounces quadratically.
     - It bends cubically.
     - X-Intercepts: To find x-intercepts, set y=0.
     - Y-Intercepts: To find y-intercepts, set x=0.

Complex Conjugates

• Complex Conjugate Theorem
     - If (x + bi) is a factor of a polynomial, then its conjugate (x - bi) is also a factor.

End-Behaviors of Polynomial Functions

• End-Behaviors:
     - If the leading coefficients are the same, end-behaviors are the same:
       - For instance, both go up or down at the extremes.
     - If they have opposite signs, they go in opposite directions:
       - One will go up while the other goes down at infinity.

Rational Functions

• General Form:
     $f(x) = rac{p(x)}{q(x)}$
     where p(x) is a numerator polynomial and q(x) is a non-zero polynomial.
     - Vertical Asymptotes: Occur when the denominator equals zero but not canceled by a factor in the numerator.
     - Horizontal Asymptotes:
       - Case I: If degree of p < degree of q, then the horizontal asymptote is $y = 0$.      - Case II: If degree of p = degree of q, then $y = rac{a}{b}$, where a and b are leading coefficients of p and q respectively.      - Case III: If degree of p > degree of q, no horizontal asymptote exists; rather, it has a slant or oblique asymptote.
     - Finding Slant/Oblique Asymptotes: Use long division when degree of numerator is one greater than degree of denominator.

• Holes: Occur where a common factor cancels. Location found by substituting the factor's zero into the simplified expression.

Sequences and Series

• Linear Sequences: Have the same differences between terms.
     - Common difference d:
       - $d = a_n - a_{n-1}$
     - General form: $a_n = a_1 + (n - 1)d$

• Exponential Sequences: Have common ratios between consecutive terms.
     - General form: $a_n = a_1 r^{n-1}$
     - Recursive definition: $a_n = a_{n-1} imes r$

Exponential and Logarithmic Functions

• Exponential Function:
     $f(x) = a imes b^x$ where b > 0.
     - Growth for b > 1.
     - Decay for 0 < b < 1.

• Logarithmic Functions:
     - Logarithm Base b properties:
       - $ext{log}_b(mn) = ext{log}_b(m) + ext{log}_b(n)$ (Product Property)
       - $ext{log}_b rac{m}{n} = ext{log}_b(m) - ext{log}_b(n)$ (Quotient Property)
       - $ext{log}_b(m^k) = k imes ext{log}_b(m)$ (Power Property)
     - Natural Logarithm properties (base e):
       - $ext{ln}(1) = 0$
       - $ext{ln}(e) = 1$

One-to-One Functions and Inverses

• A one-to-one function possesses an inverse.

• For any function $f$, if $f(a) = b$, then the inverse $f^{-1}(b) = a$.

• Composition: The composition of a function and its inverse results in the identity;
     $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
     - The domains and ranges of exponential functions and logarithmic functions are opposites of one another.

Parent Functions

• Understand the characteristics of different parent functions:
     - Linear: $y = x$
     - Quadratic: $y = x^2$
     - Square Root: $y = ext{√}x$
     - Logarithmic: $y = ext{log}_2(x)$
     - Cubic: $y = x^3$
     - Exponential: $y = 2^x$
     - Natural Exponential: $y = e^x$
     - Natural Logarithm: $y = ext{ln}(x)$.

Trigonometric Functions

• Basic definitions and relationships:
     - Sine: $ext{sin}(x)$
     - Cosine: $ext{cos}(x)$
     - Tangent: $ext{tan}(x)$
     - Cosecant: $ext{csc}(x) = rac{1}{ ext{sin}(x)}$
     - Secant: $ext{sec}(x) = rac{1}{ ext{cos}(x)}$
     - Cotangent: $ext{cot}(x) = rac{1}{ ext{tan}(x)}$

Key Trigonometric Identities

• Reciprocal Functions:
     - $ext{csc}(x) = rac{1}{ ext{sin}(x)}$
     - $ext{sec}(x) = rac{1}{ ext{cos}(x)}$

• Pythagorean Identity:
     - $ext{sin}^2(x) + ext{cos}^2(x) = 1$

Graphical Characteristics

• Amplitude and Periodicity:
     - For sine and cosine functions, amplitude is the height from the midline to the peak.
     - The period is the distance along the x-axis needed for the function to repeat.
     - Phase Shift: Horizontal shift of the graph.
     - Vertical Shift: Shifting up or down on the y-axis.

Polar Coordinates

• Conversion:
     - From polar to rectangular: $x = r imes ext{cos}( heta)$, $y = r imes ext{sin}( heta)$.
     - From rectangular to polar: $r = ext{√}(x^2 + y^2)$ and $heta = ext{tan}^{-1}( rac{y}{x})$.

Spiral and Cardioid Functions

• Spirals: Given a positive real number $a$, describe a spiral with increasing radius.
     - Domain: $[0, 2 ext{π}]$

• Cardioids: Described by polar equations
     - $r = a imes (1 + ext{cos}( heta))$
     - Domain: $[0, 2 ext{π}]$