Mr. Valsa's Math Page
  • Home
  • AP Calculus AB
    • Course Intro
    • Unit 01: Limits & Continuity >
      • 1.1 Intro to Calculus
      • 1.2 Defining Limits & Using Limit Notation
      • 1.3 Estimating Limit Values from a Graph
      • 1.4 Estimating Limit Values from a Table
      • 1.5 Determining Limits Using Algebraic Properties
      • 1.6 Determining Limits Using Algebraic Manipulation
      • 1.7 Selecting Procedures for Determining Limits
      • 1.8 Squeeze Theorem
      • 1.9 Multiple Representations of Limits
      • 1.10 Exploring Types of Discontinuities
      • 1.11 Defining Continuity
      • 1.12 Confirming Continuity over an Interval
      • 1.13 Removing Discontinuity
      • 1.14 Infinite Limits & Vertical Asymptotes
      • 1.15 Connecting Limits at Infinity & Horizontal Asymptotes
      • 1.16 Intermediate Values Theorem (IVT)
    • Unit 02: Intro to the Derivative >
      • 2.1 Defining Average & Instantaneous Rates of Change
      • 2.2 Defining the Derivative of a Function
      • 2.3 Estimating the Derivative at a Point
      • 2.4 Determining When Derivatives Exist
      • 2.5 Applying the Power Rule
      • 2.6 Derivative Rules
      • 2.7: Derivatives of sin, cos, e^x, ln(x)
      • 2.8 Product Rule
      • 2.9 Quotient Rule
      • 2.10 Derivative of tan/cot/sec/csc
    • Unit 03: Advanced Differentiation >
      • 3.1 The Chain Rule
      • 3.2 Implicit Differentiation
      • 3.3 Inverse Function Derivatives
      • 3.4 Inverse Trig Derivatives
      • 3.5 Selecting Procedures for Calculating Derivatives
      • 3.6 Higher Order Derivatives
    • Unit 04: Contextual Applications of Derivatives >
      • 4.1 The Meaning of the Derivative in Context
      • 4.2 Straight Line Motion
      • 4.3 Rates of Change in Applied Contexts
      • 4.4 Introduction to Related Rates
      • 4.5 Solving Related Rates Problems
      • 4.6 Approximating Values using Linearization
      • 4.7 L'Hospital's Rule
    • Unit 05: Analytical Applications >
      • 5.1 Using the Mean Values Theorem (MVT)
      • 5.2 Extreme Values Theorem (EVT)
      • 5.3 Determining where a function is increasing/decreasing
      • 5.4 First derivative test for extrema
      • 5.5 Candidates Test for Extreme Values
      • 5.6 Determining Concavity of a Function
      • 5.7 Second Derivative Test for Extrema
      • 5.8 Sketching Functions & their Derivatives
      • 5.9 Connecting a Function and it's 1st/2nd Derivative
      • 5.10 Intro to Optimization Problems
      • 5.11 Solving Optimization Problems
      • 5.12 Exploring Implicit Relationships
    • Unit 06: Integration & Accumulation of Change >
      • 6.1 Exploring Accumulations of Change
      • 6.2 Approximating Areas with Riemann Sums
      • 6.3 Riemann Sums/Summation Notation/Definite Integrals Notation/
      • 6.4 FTC (Fundamental Theorem of Calculus) & Accumulation
      • 6.5 Interpreting the Behavior of Accumulation Functions
      • 6.6 Applying Properties of Definite Integrals
      • 6.7 FTC and Definite Integrals
      • 6.8 Anti-Derivatives - Basic Rules & Notation
      • 6.9 Integration Using Substitution
      • 6.10 Integration by Long Division & Completing the Square
      • 6.14 - Selecting Techniques of Anti-Differentiation
    • Unit 07: Differential Equations >
      • 7.1 Modeling Situations with Differential Equations
      • 7.2 Verifying Solutions to Differential Equations
      • 7.3 Sketching Slope Fields
      • 7.4 Reasoning with Slope Fields
      • 7.6 Finding General Solutions using Separation of Variables
      • 7.7 Finding Particular Solutions using Initial Conditions
      • 7.8 Exponential Models with Differential Equations
    • Unit 08: Applications of Integration
  • PreCalculus
    • Course Intro
    • Unit 01 - Function Analysis >
      • 1.1 Set & Interval Notation
      • 1.2 Function Review
      • 1.3 Domain & Range
      • 1.4 End Behavior & Extrema
      • 1.5 Inc/Dec/Const
      • 1.6 Symmetry
      • 1.7 Function Transformations
      • 1.8 Function Operations
      • 1.9 Inverses
    • Unit 02: Polynomial & Rational Functions >
      • 2.1 Solving Polynomials
      • 2.2 Classifying Polynomials
      • 2.3 Complex Numbers
      • 2.4 End Behavior
      • 2.5 Writing Equations of Polynomials
      • 2.6 Solving Rational Equations
      • 2.7 Rational Inequalities
      • 2.8 Types of Discontinuity
      • 2.9 Graphing Rational Functions
    • Unit 03 - Conic Sections >
      • 3.1 Intro to Conic Sections
      • 3.2 Circles
      • 3.3 Ellipses
      • 3.4 Hyperbolas
      • 3.5 Parabolas
    • Unit 04 - Exponents & Logarithms >
      • 4.1 Exponents & Logarithms
  • Integrated Math 3
    • Course Intro
    • Unit 01: Analytic Geometry >
      • 1.1 Graphing Lines
      • 1.2 Parallel & Perpendicular
      • 1.3 Distance & Pythagorean Theorem
      • 1.4 Classifying Quadrilaterals
    • Unit 02: Geometric Proofs & Modeling >
      • 2.1 Surface Area & Volume
      • 2.2 Cross Sections
    • Unit 03: Circles & Parabolas >
      • 3.1 Circles Review
      • 3.2 Arc Length & Sector Area
      • 3.3 Writing & Graphing Circles
      • 3.4 Completing the Square
      • 3.5 Parabolas
    • Unit 04 - Representing Functions >
      • 4.1 - Key Features
      • 4-2 Even & Odd Symmetry
      • 4.3 Graphing Review
      • 4.4 Absolute Value Equations
      • 4.5 Absolute Value Functions Cont.
      • 4.6 Piecewise Functions
  • Math Team
  • Contact

​2.9 Graphing Rational Functions

2.9_notes_part_1_-_precalc.pdf
File Size: 432 kb
File Type: pdf
Download File
2.9_completed_notes_part_1_-_precalc.pdf
File Size: 211 kb
File Type: pdf
Download File
2.9_notes_part_2_-_precalc.pdf
File Size: 271 kb
File Type: pdf
Download File
2.9_completed_notes_part_2_-_precalc.pdf
File Size: 229 kb
File Type: pdf
Download File
2.9_toolkit_-_precalc.pdf
File Size: 148 kb
File Type: pdf
Download File
2.9_completed_toolkit_-_precalc.pdf
File Size: 106 kb
File Type: pdf
Download File
Proudly powered by Weebly
  • Home
  • AP Calculus AB
    • Course Intro
    • Unit 01: Limits & Continuity >
      • 1.1 Intro to Calculus
      • 1.2 Defining Limits & Using Limit Notation
      • 1.3 Estimating Limit Values from a Graph
      • 1.4 Estimating Limit Values from a Table
      • 1.5 Determining Limits Using Algebraic Properties
      • 1.6 Determining Limits Using Algebraic Manipulation
      • 1.7 Selecting Procedures for Determining Limits
      • 1.8 Squeeze Theorem
      • 1.9 Multiple Representations of Limits
      • 1.10 Exploring Types of Discontinuities
      • 1.11 Defining Continuity
      • 1.12 Confirming Continuity over an Interval
      • 1.13 Removing Discontinuity
      • 1.14 Infinite Limits & Vertical Asymptotes
      • 1.15 Connecting Limits at Infinity & Horizontal Asymptotes
      • 1.16 Intermediate Values Theorem (IVT)
    • Unit 02: Intro to the Derivative >
      • 2.1 Defining Average & Instantaneous Rates of Change
      • 2.2 Defining the Derivative of a Function
      • 2.3 Estimating the Derivative at a Point
      • 2.4 Determining When Derivatives Exist
      • 2.5 Applying the Power Rule
      • 2.6 Derivative Rules
      • 2.7: Derivatives of sin, cos, e^x, ln(x)
      • 2.8 Product Rule
      • 2.9 Quotient Rule
      • 2.10 Derivative of tan/cot/sec/csc
    • Unit 03: Advanced Differentiation >
      • 3.1 The Chain Rule
      • 3.2 Implicit Differentiation
      • 3.3 Inverse Function Derivatives
      • 3.4 Inverse Trig Derivatives
      • 3.5 Selecting Procedures for Calculating Derivatives
      • 3.6 Higher Order Derivatives
    • Unit 04: Contextual Applications of Derivatives >
      • 4.1 The Meaning of the Derivative in Context
      • 4.2 Straight Line Motion
      • 4.3 Rates of Change in Applied Contexts
      • 4.4 Introduction to Related Rates
      • 4.5 Solving Related Rates Problems
      • 4.6 Approximating Values using Linearization
      • 4.7 L'Hospital's Rule
    • Unit 05: Analytical Applications >
      • 5.1 Using the Mean Values Theorem (MVT)
      • 5.2 Extreme Values Theorem (EVT)
      • 5.3 Determining where a function is increasing/decreasing
      • 5.4 First derivative test for extrema
      • 5.5 Candidates Test for Extreme Values
      • 5.6 Determining Concavity of a Function
      • 5.7 Second Derivative Test for Extrema
      • 5.8 Sketching Functions & their Derivatives
      • 5.9 Connecting a Function and it's 1st/2nd Derivative
      • 5.10 Intro to Optimization Problems
      • 5.11 Solving Optimization Problems
      • 5.12 Exploring Implicit Relationships
    • Unit 06: Integration & Accumulation of Change >
      • 6.1 Exploring Accumulations of Change
      • 6.2 Approximating Areas with Riemann Sums
      • 6.3 Riemann Sums/Summation Notation/Definite Integrals Notation/
      • 6.4 FTC (Fundamental Theorem of Calculus) & Accumulation
      • 6.5 Interpreting the Behavior of Accumulation Functions
      • 6.6 Applying Properties of Definite Integrals
      • 6.7 FTC and Definite Integrals
      • 6.8 Anti-Derivatives - Basic Rules & Notation
      • 6.9 Integration Using Substitution
      • 6.10 Integration by Long Division & Completing the Square
      • 6.14 - Selecting Techniques of Anti-Differentiation
    • Unit 07: Differential Equations >
      • 7.1 Modeling Situations with Differential Equations
      • 7.2 Verifying Solutions to Differential Equations
      • 7.3 Sketching Slope Fields
      • 7.4 Reasoning with Slope Fields
      • 7.6 Finding General Solutions using Separation of Variables
      • 7.7 Finding Particular Solutions using Initial Conditions
      • 7.8 Exponential Models with Differential Equations
    • Unit 08: Applications of Integration
  • PreCalculus
    • Course Intro
    • Unit 01 - Function Analysis >
      • 1.1 Set & Interval Notation
      • 1.2 Function Review
      • 1.3 Domain & Range
      • 1.4 End Behavior & Extrema
      • 1.5 Inc/Dec/Const
      • 1.6 Symmetry
      • 1.7 Function Transformations
      • 1.8 Function Operations
      • 1.9 Inverses
    • Unit 02: Polynomial & Rational Functions >
      • 2.1 Solving Polynomials
      • 2.2 Classifying Polynomials
      • 2.3 Complex Numbers
      • 2.4 End Behavior
      • 2.5 Writing Equations of Polynomials
      • 2.6 Solving Rational Equations
      • 2.7 Rational Inequalities
      • 2.8 Types of Discontinuity
      • 2.9 Graphing Rational Functions
    • Unit 03 - Conic Sections >
      • 3.1 Intro to Conic Sections
      • 3.2 Circles
      • 3.3 Ellipses
      • 3.4 Hyperbolas
      • 3.5 Parabolas
    • Unit 04 - Exponents & Logarithms >
      • 4.1 Exponents & Logarithms
  • Integrated Math 3
    • Course Intro
    • Unit 01: Analytic Geometry >
      • 1.1 Graphing Lines
      • 1.2 Parallel & Perpendicular
      • 1.3 Distance & Pythagorean Theorem
      • 1.4 Classifying Quadrilaterals
    • Unit 02: Geometric Proofs & Modeling >
      • 2.1 Surface Area & Volume
      • 2.2 Cross Sections
    • Unit 03: Circles & Parabolas >
      • 3.1 Circles Review
      • 3.2 Arc Length & Sector Area
      • 3.3 Writing & Graphing Circles
      • 3.4 Completing the Square
      • 3.5 Parabolas
    • Unit 04 - Representing Functions >
      • 4.1 - Key Features
      • 4-2 Even & Odd Symmetry
      • 4.3 Graphing Review
      • 4.4 Absolute Value Equations
      • 4.5 Absolute Value Functions Cont.
      • 4.6 Piecewise Functions
  • Math Team
  • Contact