Mr. Valsa's Math Page
  • Home
  • AP Calculus AB
    • Course Intro
    • Semester 1 Review Material
    • 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 5 Review Material
    • 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 06 Review Material
    • 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 07 Review Material
    • Unit 08: Applications of Integration >
      • 8.1 Average Value of a Function
      • 8.2 Connecting Position, Velocity & Acceleration
      • 8.3 Applied Contexts with Definite Integrals
      • 8.4 Area Between Curves in terms of X
      • 8.5 Area Between Curves in terms of Y
      • 8.6 Area Between Curves that Intersect at More Than Two Points
      • 8.7 Cross Sectional Volume: Squares/Rectangles
      • 8.8 Cross Sectional Volume: Triangles/Semicircles
      • 8.9 Volume with Disc Method: Revolving Around the X or Y Axis
      • 8.10 Volume with Disc Method: Revolving Around Other Axis
      • 8.11 Volume with Washer Method: Revolving Around the X or Y Axis
      • 8.12 Volume with Washer Method: Revolving Around Other Axes
    • AP Review >
      • Unit Recaps
      • Problem Sets
      • Question Types
      • Free Response Questions
  • 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
      • 1.10 Piecewise Functions
      • Unit 01 Review Material
    • 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 02 Review Material
    • Unit 03: Conic Sections >
      • 3.1 Intro to Conic Sections
      • 3.2 Circles
      • 3.3 Ellipses
      • 3.4 Hyperbolas
      • 3.5 Parabolas
      • 3.6 Systems of Conic Sections
      • Unit 03 Review
    • Unit 04: Exponents & Logarithms >
      • 4.1 Exponents & Logarithms
      • 4.2 Evaluating Logarithms
      • 4.3 Solving Exponential & Log Equations
      • 4.4 Growth & Decay Models
      • 4.5 Properties of Logarithms
      • 4.6 Graphing Exponentials
      • Unit 04 Review Material
    • Unit 05: Sequences & Series >
      • 5.1 Arithmetic & Geometric Sequences
      • 5.2 Arithmetic Series
      • 5.3 Finite Geometric Series
      • 5.4 Infinite Geometric Series
      • 5.6 Sigma Notation
      • Unit 05 Review Material
    • Unit 06: Right Triangle Trigonometry >
      • 6.1 Right Angle Trigonometry
      • 6.2 Applications of Trigonometry
      • 6.3 Standard Position, Degrees, Radians
      • 6.4 Coterminal & Reference Angles
      • 6.5 Evaluating Trig Functions
      • 6.6 The Unit Circle
      • 6.7 Inverse Trig
      • 6.8 Law of Sines/Cosines
      • 6.9 Applications Part 2
      • Unit 06 Review Material
    • Unit 07: Graphing Trigonometric Functions >
      • 7.1 Graphing Basic Trig
      • 7.2 Graphing Trig Functions
      • 7.3 Trig Graphs w/ Phase Shifts
      • 7.4 Writing Equations of Sine/Cosine Graphs
      • 7.5 Modeling with Trig
      • 7.6 Graphing Tangent
      • 7.7 Writing Tangent Equations
      • Unit 07 Review Material
    • Unit 08: Trigonometric Identities >
      • 8.1 Complex Fractions
      • 8.2 Simplifying Trig Expressions
      • 8.3 Verifying Identities
      • 8.4 Sum & Difference Identities
      • 8.5 Half & Double Angle Identities
      • 8.6 Trigonometric Equations
      • Unit 08 Review Material
    • Unit 09: Polar Coordinates >
      • 9.1 Intro to Polar
      • 9.2 Polar Equations
      • 9.3 Basic Polar Graphs
      • Unit 09 Review Material
    • Unit 10: Vectors >
      • 10.1 Intro to Vectors
      • 10.2 Vector Addition/Subtraction
      • 10.3 Unit Vectors
      • 10.4 Vector Applications
      • 10.5 Dot Product
      • 10.6 Cross Product
      • Unit 10 Review Material
    • Unit 11: Intro to Calculus >
      • 11.1 Graphical Limits
      • 11.2 Algebraic Limits
      • 11.3 Continuity
    • Semester 2 Review
  • Integrated Math 3
    • Course Intro
    • Semester 1 Review Material
    • Unit 01: Analytic Geometry >
      • 1.1 Graphing Lines
      • 1.2 Parallel & Perpendicular
      • 1.3 Distance & Pythagorean Theorem
      • 1.4 Classifying Quadrilaterals
      • Unit 01 Review
    • Unit 02: Geometric Proofs & Modeling >
      • 2.1 Surface Area & Volume
      • 2.2 Density
      • 2.3 Cross Sections
      • Unit 02 Review
    • 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 03 Review
    • 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
      • Unit 04 Review Material
    • Unit 05: Trigonometry >
      • 5.0 Intro to Trig
      • 5.1 Trig Applications
      • 5.2 Degrees & Radians
      • 5.3 Special Right Triangles
      • 5.4 Evaluating Trig Functions
      • 5.5 Reference Triangles
      • Unit 05 Review Material
    • Unit 06: Graphing Trig >
      • 6.0 Intro to Trig Graphs
      • 6.1 Vertical Shifts
      • 6.2 Phase Shifts
      • 6.3 Writing Trig Equations
      • Unit 06 Review Material
    • Unit 07: Polynomial Functions >
      • 7.0 Operations & Classifying Polynomials
      • 7.1 Factoring
      • 7.2 Solving by Factoring
      • 7.3 Solving Simultaneous Functions
      • 7.4 Exploring Changes in Graphs
      • 7.5 Key Features
      • 7.6 End Behavior
      • 7.7 Graphing Polynomials
      • 7.8 Factor Theorem
      • 7.9 Inverses
      • Unit 07 Review Material
    • Unit 08: Rational Functions >
      • 8.0 Simplifying Rational Expressions
      • 8.1 Adding/Subtracting Rational Expressions
      • 8.2 Solving Rational Equations
      • 8.3 Key Features
      • 8.4 Inverses
      • Unit 08 Review Material
    • Unit 09: Log & Exponential Functions >
      • 9.0 Exponent Rules
      • 9.1 Common Bases
      • 9.2 Uncommon Bases
      • 9.3 Evaluating & Rewriting
      • 9.4 Natural Logarithms
      • 9.5 Inverses
      • 9.6 Exponential Graphs
      • 9.7 Transformations of Exponentials
      • 9.8 Graphing Logarithms
      • 9.9 Applications
      • Unit 09 Review Material
    • Unit 10: Inferential Statistics >
      • 10.0 Types of Studies
      • 10.1 Sampling Methods
      • 10.2 Sampling Technique Summary
      • 10.3 Measures of Center
      • 10.4 Histograms & Shape
      • 10.5 Precision vs. Accuracy
      • 10.6 Empirical Rule & Z-Scores
      • Unit 10 Review Material
    • Unit 11: Matrices >
      • 11.0 Adding & Subtracting Matrices
      • 11.1 Multiplying Matrices
      • 11.2 Determinant & Inverse
      • 11.3 Solving Matrices
      • 11.4 Matrix Applications
      • Unit 11 Review Material
    • SAT Practice
    • Semester 2 Review
  • Math Team
  • Contact

​2.9 Graphing Rational Functions

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  • Home
  • AP Calculus AB
    • Course Intro
    • Semester 1 Review Material
    • 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 5 Review Material
    • 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 06 Review Material
    • 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 07 Review Material
    • Unit 08: Applications of Integration >
      • 8.1 Average Value of a Function
      • 8.2 Connecting Position, Velocity & Acceleration
      • 8.3 Applied Contexts with Definite Integrals
      • 8.4 Area Between Curves in terms of X
      • 8.5 Area Between Curves in terms of Y
      • 8.6 Area Between Curves that Intersect at More Than Two Points
      • 8.7 Cross Sectional Volume: Squares/Rectangles
      • 8.8 Cross Sectional Volume: Triangles/Semicircles
      • 8.9 Volume with Disc Method: Revolving Around the X or Y Axis
      • 8.10 Volume with Disc Method: Revolving Around Other Axis
      • 8.11 Volume with Washer Method: Revolving Around the X or Y Axis
      • 8.12 Volume with Washer Method: Revolving Around Other Axes
    • AP Review >
      • Unit Recaps
      • Problem Sets
      • Question Types
      • Free Response Questions
  • 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
      • 1.10 Piecewise Functions
      • Unit 01 Review Material
    • 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 02 Review Material
    • Unit 03: Conic Sections >
      • 3.1 Intro to Conic Sections
      • 3.2 Circles
      • 3.3 Ellipses
      • 3.4 Hyperbolas
      • 3.5 Parabolas
      • 3.6 Systems of Conic Sections
      • Unit 03 Review
    • Unit 04: Exponents & Logarithms >
      • 4.1 Exponents & Logarithms
      • 4.2 Evaluating Logarithms
      • 4.3 Solving Exponential & Log Equations
      • 4.4 Growth & Decay Models
      • 4.5 Properties of Logarithms
      • 4.6 Graphing Exponentials
      • Unit 04 Review Material
    • Unit 05: Sequences & Series >
      • 5.1 Arithmetic & Geometric Sequences
      • 5.2 Arithmetic Series
      • 5.3 Finite Geometric Series
      • 5.4 Infinite Geometric Series
      • 5.6 Sigma Notation
      • Unit 05 Review Material
    • Unit 06: Right Triangle Trigonometry >
      • 6.1 Right Angle Trigonometry
      • 6.2 Applications of Trigonometry
      • 6.3 Standard Position, Degrees, Radians
      • 6.4 Coterminal & Reference Angles
      • 6.5 Evaluating Trig Functions
      • 6.6 The Unit Circle
      • 6.7 Inverse Trig
      • 6.8 Law of Sines/Cosines
      • 6.9 Applications Part 2
      • Unit 06 Review Material
    • Unit 07: Graphing Trigonometric Functions >
      • 7.1 Graphing Basic Trig
      • 7.2 Graphing Trig Functions
      • 7.3 Trig Graphs w/ Phase Shifts
      • 7.4 Writing Equations of Sine/Cosine Graphs
      • 7.5 Modeling with Trig
      • 7.6 Graphing Tangent
      • 7.7 Writing Tangent Equations
      • Unit 07 Review Material
    • Unit 08: Trigonometric Identities >
      • 8.1 Complex Fractions
      • 8.2 Simplifying Trig Expressions
      • 8.3 Verifying Identities
      • 8.4 Sum & Difference Identities
      • 8.5 Half & Double Angle Identities
      • 8.6 Trigonometric Equations
      • Unit 08 Review Material
    • Unit 09: Polar Coordinates >
      • 9.1 Intro to Polar
      • 9.2 Polar Equations
      • 9.3 Basic Polar Graphs
      • Unit 09 Review Material
    • Unit 10: Vectors >
      • 10.1 Intro to Vectors
      • 10.2 Vector Addition/Subtraction
      • 10.3 Unit Vectors
      • 10.4 Vector Applications
      • 10.5 Dot Product
      • 10.6 Cross Product
      • Unit 10 Review Material
    • Unit 11: Intro to Calculus >
      • 11.1 Graphical Limits
      • 11.2 Algebraic Limits
      • 11.3 Continuity
    • Semester 2 Review
  • Integrated Math 3
    • Course Intro
    • Semester 1 Review Material
    • Unit 01: Analytic Geometry >
      • 1.1 Graphing Lines
      • 1.2 Parallel & Perpendicular
      • 1.3 Distance & Pythagorean Theorem
      • 1.4 Classifying Quadrilaterals
      • Unit 01 Review
    • Unit 02: Geometric Proofs & Modeling >
      • 2.1 Surface Area & Volume
      • 2.2 Density
      • 2.3 Cross Sections
      • Unit 02 Review
    • 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 03 Review
    • 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
      • Unit 04 Review Material
    • Unit 05: Trigonometry >
      • 5.0 Intro to Trig
      • 5.1 Trig Applications
      • 5.2 Degrees & Radians
      • 5.3 Special Right Triangles
      • 5.4 Evaluating Trig Functions
      • 5.5 Reference Triangles
      • Unit 05 Review Material
    • Unit 06: Graphing Trig >
      • 6.0 Intro to Trig Graphs
      • 6.1 Vertical Shifts
      • 6.2 Phase Shifts
      • 6.3 Writing Trig Equations
      • Unit 06 Review Material
    • Unit 07: Polynomial Functions >
      • 7.0 Operations & Classifying Polynomials
      • 7.1 Factoring
      • 7.2 Solving by Factoring
      • 7.3 Solving Simultaneous Functions
      • 7.4 Exploring Changes in Graphs
      • 7.5 Key Features
      • 7.6 End Behavior
      • 7.7 Graphing Polynomials
      • 7.8 Factor Theorem
      • 7.9 Inverses
      • Unit 07 Review Material
    • Unit 08: Rational Functions >
      • 8.0 Simplifying Rational Expressions
      • 8.1 Adding/Subtracting Rational Expressions
      • 8.2 Solving Rational Equations
      • 8.3 Key Features
      • 8.4 Inverses
      • Unit 08 Review Material
    • Unit 09: Log & Exponential Functions >
      • 9.0 Exponent Rules
      • 9.1 Common Bases
      • 9.2 Uncommon Bases
      • 9.3 Evaluating & Rewriting
      • 9.4 Natural Logarithms
      • 9.5 Inverses
      • 9.6 Exponential Graphs
      • 9.7 Transformations of Exponentials
      • 9.8 Graphing Logarithms
      • 9.9 Applications
      • Unit 09 Review Material
    • Unit 10: Inferential Statistics >
      • 10.0 Types of Studies
      • 10.1 Sampling Methods
      • 10.2 Sampling Technique Summary
      • 10.3 Measures of Center
      • 10.4 Histograms & Shape
      • 10.5 Precision vs. Accuracy
      • 10.6 Empirical Rule & Z-Scores
      • Unit 10 Review Material
    • Unit 11: Matrices >
      • 11.0 Adding & Subtracting Matrices
      • 11.1 Multiplying Matrices
      • 11.2 Determinant & Inverse
      • 11.3 Solving Matrices
      • 11.4 Matrix Applications
      • Unit 11 Review Material
    • SAT Practice
    • Semester 2 Review
  • Math Team
  • Contact