AP PreCalc

1. Functions

• Definition of a Function: Relation between inputs and outputs, where each input has exactly one output.

• Domain and Range:
  
  

  • Domain: Set of possible input values.

  • Range: Set of possible output values.
    
    

Types of Functions

• Linear Functions:
  
  

  • Slope-intercept form: $y = mx + b$

  • Point-slope form: $y - y_1 = m(x - x_1)$

• Quadratic Functions:
  
  

  • Standard form: $y = ax^2 + bx + c$

  • Vertex form: $y = a(x-h)^2 + k$.

  • Graphing parabolas; discriminants for determining roots.

• Polynomial Functions:
  
  

  • Degree and leading coefficient determine the end behavior.

• Rational Functions:
  
  

  • Asymptotes: Vertical (undefined points), horizontal (end behavior).

• Exponential Functions:
  
  

  • Describes growth or decay; applications in real-world scenarios.

• Logarithmic Functions:
  
  

  • Inverses of exponential functions; properties and graphs.

  • Change of base formula for conversion.   

Function Operations

• Addition, subtraction, multiplication, division of functions

• Inverse functions: A function that reverses another.

• Composition of functions: $f(g(x))$, understanding the output of one function becomes the input to another.

2. Trigonometry

• Basic Trigonometric Ratios:
  
  

  • Sine: $\frac{\text{opposite}}{\text{hypotenuse}}$

  • Cosine: $\frac{\text{adjacent}}{\text{hypotenuse}}$

  • Tangent: $\frac{\text{opposite}}{\text{adjacent}}$

• The Unit Circle:
  
  

  • Angles measured in radians and degrees.

  • Key angles: 0, \frac{\frac{\text{π}}{6}}, $\frac{\text{π}}{4}$, \frac{\frac{\text{π}}{3}}, $\text{π}$, and their coordinates.

• Trigonometric Identities:
  
  

  • Pythagorean identities: $\text{sin}^2(x) + \text{cos}^2(x) = 1$.

  • Angle addition and subtraction formulas; double-angle formulas.

• Inverse Trigonometric Functions:
  
  

  • Functions that return angles based on trigonometric ratios.

• Applications of Trigonometry:
  
  

  • Law of Sines and Law of Cosines for solving triangles.

3. Complex Numbers

• Definition and Operations:
  
  

  • Form: $a + bi$, where $i = \text{sqrt}(-1)$.

  • Operations include addition, subtraction, multiplication, division.

• Polar Form:
  
  

  • Converting to polar coordinates; using magnitudes and angles.

• De Moivre's Theorem:
  
  

  • Finding powers and roots of complex numbers in polar form.

4. Polynomial and Rational Functions

• Characteristics of Polynomials:
  
  

  • Degree determines the number of zeros.

  • Leading coefficient affects end behavior.

• Graphing Polynomial Functions:
  
  

  • Finding zeros and using them to sketch graphs.

• Rational Functions:
  
  

  • Asymptotic behavior and how to graph rational expressions.
    
    

5. Exponential and Logarithmic Functions

• Properties of Exponential Functions:
  
  

  • Graphs and characteristics of growth/decay.

• Laws of Logarithms:
  
  

  • Product, quotient, and power rules.

  • Solving logarithmic equations.

• Applications in Real-World Scenarios:
  
  

  • Situations involving exponential growth, decay, financial models, and population growth.

6. Sequences and Series

• Arithmetic and Geometric Sequences:
  
  

  • Finding general terms and sums for both types.

• Summation Notation:
  
  

  • Understanding how to represent sums compactly.

• Binomial Theorem:
  
  

  • Expanding binomials using coefficients defined by combinations.

7. Analytic Geometry

• Conic Sections:
  
  

  • Characteristics, equations, and applications of circles, ellipses, parabolas, and hyperbolas.

8. Limits

• Understanding the Concept of a Limit:
  
  

  • Definitions and approaches for finding limits graphically and numerically.

  • Techniques for evaluating limits, including substitution and algebraic simplification.
    
    

9. Vectors and Matrices

• Vectors:
  
  

  • Operations: addition, subtraction, and scalar multiplication; understanding directions and magnitudes.

• Dot Product and Cross Product:
  
  

  • Applications in physics and geometry.

• Matrices:
  
  

  • Operations: addition, subtraction, multiplication; applications to systems of equations, transformations, and data representation.

Advanced Topics in Precalculus

• Piecewise Functions:
  
  

  • Defined by multiple sub-functions, each applicable to a specific interval.

  • Graphs can show discontinuities or jumps at the boundaries.

• Transformation of Functions:
  
  

  • Includes translations, reflections, stretches, and compressions.

  • General form: $y = a(f(b(x - h))) + k$, where $a$ controls vertical scaling, $b$ horizontal scaling, $h$ horizontal translation, and $k$ vertical translation.

• Implicit Functions:
  
  

  • Not expressed in standard $y = f(x)$ form. Example: $x^2 + y^2 = r^2$ describes a circle.

• Parametric Equations:
  
  

  • Represent graphs with two variables defined by a time parameter $t$. Example: $x(t) = r \times \text{cos}(t), y(t) = r \times \text{sin}(t)$ for a circle.

• Polar Coordinates:
  
  

  • Uses radial distance and angle, rather than traditional rectangular coordinates. Example: Point defined as $(r, \theta)$. Conversion to Cartesian: $x = r \times \text{cos}(\theta), y = r \times \text{sin}(\theta)$.

• Conic Sections:
  
  

  • More detailed look into hyperbolas, ellipses, and parabolas.

  • Focus on focal points, directrix, eccentricity, and the standard forms derived from the general second-degree equation.

• Sequences Variations:
  
  

  • Exploring harmonic and Fibonacci sequences in further depth.

  • General formulas and recursive definitions to generate terms.

• Complex Number Applications:
  
  

  • Connection between complex numbers and trigonometric forms: $z = re^{i\theta}$ and how this relates to exponential growth and decay.

• Binomial Series:
  
  

  • Generalized version of the Binomial Theorem for non-integer exponents.

  • Applications of converging series and regrouping terms for approximation.

• Limits and Continuity:
  
  

  • Methods for evaluating limits analytically and graphically, including L'Hôpital's rule for indeterminate forms.

  • Understanding of continuity, differentiability, and the difference in behavior in piecewise functions.