Pre-Calculus

  • Modeling and Equation Solving - Algebraic Models and Geometric Models
  • Zero Factor Property, Grapher Failure and Hidden Behaviour
  • Function Definition and Notation
  • Domain and Range of a Function
  • Continuity
  • Increasing and Decreasing Functions
  • Boundedness, Local and Absolute Extrema, Symmetry, Asymptotes, End Behaviour
  • Twelve Basic Functions
  • Combining Functions Algebraically and Composition of Functions
  • Relations and Implicitly Defined Functions
  • Relations Defined Parametrically
  • Inverse Relations and Inverse Functions
  • Vertical and Horizontal Translations, Reflections Across Axes, Vertical and Horizontal Stretches and Shrinks, Combining Transformations
  • Forming Functions from Formulas
  • Functions from Graphs, Functions from Verbal Descriptions, Functions from Data
  • Polynomial Functions
  • Linear Functions and Their Graphs
  • Average Rate of Change
  • Association, Correlation, and Linear Modeling
  • Quadratic Functions and Their Graphs
  • Applications of Quadratic Functions
  • Power Functions and Variation, Graphs of Power Functions, Modeling with Power Functions
  • Monomial Functions and Their Graphs
  • End Behavior of Polynomial Functions
  • Zeros of Polynomial Functions
  • Intermediate Value Theorem
  • Modeling
  • Polynomial Division - Long Division and the Division Algorithm
  • Remainder and Factor Theorems, Synthetic Division
  • Rational Zeros Theorem
  • Upper and Lower Bounds
  • Fundamental Theorem of Algebra, Linear Factorization Theorem
  • Complex Conjugate Zeros
  • Factoring with Real Number Coefficients
  • Rational Functions and Domain of Rational Functions
  • Transformations of the Reciprocal Function
  • Limits and Asymptotes
  • Analyzing Graphs of Rational Functions
  • Exploring Relative Humidity
  • Solving Rational Equations - Extraneous Solutions
  • Exponential Functions and Their Graphs
  • Transforming Exponential Functions
  • Natural Base e
  • Logistic Functions and Their Graphs
  • Graphing Logistic Growth Functions
  • Constant Percentage Rate and Exponential Functions
  • Exponential Growth and Decay Models
  • Using Regression to Model Population
  • Inverses of Exponential Functions
  • Changing Between Logarithmic and Exponential Form
  • Evaluating Logarithmic and Exponential Expressions
  • Common Logarithms—Base 10
  • Basic Properties of Common Logarithms
  • Natural Logarithms—Base e
  • Graphs of Logarithmic Functions
  • Transforming Logarithmic Graphs
  • Measuring Sound Using Decibels
  • Properties of Logarithms
  • Exploring the Arithmetic of Logarithms - Expanding the Logarithm of a Product, Expanding the Logarithm of a Quotient.
  • Change of Base
  • Graphs of Logarithmic Functions with Base b
  • Solving Exponential Equations
  • Solving Logarithmic Equations
  • Orders of Magnitude and Logarithmic Models
  • Newton’s Law of Cooling
  • Logarithmic Re-expression
  • Degrees and Radians
  • Circular Arc Length
  • Angular and Linear Motion
  • Right Triangle Trigonometry
  • Evaluating Trigonometric Functions of 45° & Evaluating Trigonometric Functions of 30°
  • Applications of Right Triangle Trigonometry
  • Trigonometric Functions of Any Angle
  • Evaluating Trig Functions of Quadrantal Angles
  • Trigonometric Functions of Real Numbers
  • Periodic Functions
  • The 16-Point Unit Circle
  • Sine and Cosine Functions
  • Sinusoids and Transformations
  • Modeling Periodic Behavior with Sinusoids
  • Tangent and Cotangent Function
  • Secant and Cosecant Function
  • Inverse Sine, Cosine and Tangent Functions
  • Composing Trigonometric and Inverse Trigonometric Functions
  • Applications of Inverse Trigonometric Functions
  • Simple Harmonic Motion
  • Basic Trigonometric Identities
  • Pythagorean Identities
  • Cofunction Identities
  • Odd-Even Identities
  • Simplifying Trigonometric Expressions
  • Solving Trigonometric Equations
  • Proving Identities and Disproving Non-Identities
  • Identities in Calculus
  • Cosine of a Difference and Cosine of a sum
  • Sine of a Difference or Sum
  • Tangent of a Difference or Sum
  • Verifying a Sinusoid Algebraically
  • Double-Angle Identities
  • Power-Reducing Identities
  • Half-Angle Identities
  • Deriving the Law of Sines
  • Solving Triangles (AAS, ASA) and The Ambiguous Case (SSA)
  • Deriving the Law of Cosines
  • Triangle Area and Heron’s Formula


  • Two-Dimensional Vectors, Head Minus Tail (HMT) Rule for Vectors, Vector Operations
  • Unit Vector, Direction Angles
  • Applications of Vectors
  • Dot Product of Vectors, Properties of Dot Product
  • Angle Between Vectors
  • Orthogonal Vectors
  • Projecting One Vector onto Another
  • Parametric Equations and Parametric Curves
  • Eliminating the Parameter
  • Lines and Line Segments
  • Simulating Motion with a Grapher
  • Polar Coordinate System
  • Coordinate Conversion and Equation Conversion
  • Finding Distance Using Polar Coordinates
  • Polar Curves and Parametric Curves, Symmetry
  • Analyzing Polar Graphs, Rose Curves, Limaçon Curves
  • De Moivre’s Theorem and nth Roots



  • Solving system of equations - The Method of Substitution
  • Solving Systems Graphically
  • The Method of Elimination
  • Multivariate Linear Systems and Row Operations
  • Matrix Algebra
  • Systems of Inequalities in Two Variables
  • Conic Sections and Geometry of a Parabola
  • Translations of Parabolas
  • Reflective Property of a Parabola
  • Geometry of an Ellipse
  • Translations of Ellipses, Orbits and Eccentricity
  • Reflective Property of an Ellipse
  • Geometry of a Hyperbola
  • Translations of Hyperbolas
  • Eccentricity and Orbits
  • Reflective Property of a Hyperbola
  • Long-Range Navigation

  • Sequences and Series
  • Arithmetic Sequences and Partial Sums
  • Geometric Sequences and Series
  • Pascal’s Triangle
  • Binomial Theorem
  • Factorial Identities
  • Average Velocity, Instantaneous Velocity
  • The Connection to Tangent Lines
  • The Derivative
  • Distance from a Constant Velocity
  • Distance from a Changing Velocity
  • Limits at Infinity
  • The Connection to Areas
  • The Definite Integral
  • Properties of Limits
  • Limits of Continuous Functions
  • One-Sided and Two-Sided Limits
  • Limits Involving Infinity
  • Computing a Derivative from Data
  • Computing a Definite Integral from Data

At Refresh Kid tutoring sessions, we offer a comprehensive Precalculus course curriculum designed to help students build a strong foundation in the essential concepts and skills needed for success in advanced mathematics. Our goal is to ensure that students develop a deep understanding of precalculus topics and develop problem-solving abilities that will serve them well in future math courses. Here is an overview of our

  1. Functions and Graphs:
  • Understanding function notation and representation.
  • Exploring linear, quadratic, polynomial, exponential, and logarithmic functions.
  • Analyzing and graphing functions using transformations.
  • Investigating the properties of functions, including domain, range, and symmetry.
  1. Trigonometry:
  • Introducing trigonometric functions (sine, cosine, tangent) and their properties.
  • Solving right triangles and applying trigonometric ratios.
  • Understanding radian and degree measure.
  • Exploring the unit circle and trigonometric identities.
  1. Analytic Geometry:
  • Understanding the Cartesian coordinate system.
  • Exploring conic sections, including circles, ellipses, hyperbolas, and parabolas.
  • Analyzing and graphing polar equations.
  • Investigating parametric equations and their applications.
  1. Polynomial and Rational Functions:
  • Analyzing polynomial functions, including end behavior, zeros, and multiplicity.
  • Dividing polynomials using synthetic and long division.
  • Investigating rational functions, asymptotes, and solving rational equations.
  1. Exponential and Logarithmic Functions:
  • Understanding the properties and graphing of exponential and logarithmic functions.
  • Solving exponential and logarithmic equations.
  • Exploring exponential growth and decay models.
  1. Systems of Equations and Matrices:
  • Solving systems of linear equations using various methods (substitution, elimination, matrices).
  • Investigating matrix operations, including addition, subtraction, multiplication, and determinants.
  • Applying matrices to solve systems of equations and perform transformations.
  1. Sequences, Series, and Probability:
  • Understanding arithmetic and geometric sequences and series.
  • Analyzing and finding sums of finite and infinite sequences and series.
  • Exploring probability concepts and applications.

Throughout the course, students will engage in practice exercises, problem-solving tasks, and real-world applications to reinforce their understanding of the concepts covered. Our tutors provide individualized attention, clear explanations, and step-by-step guidance to help students master each topic.

We are committed to supporting students' progress and offer additional resources, practice materials, and ongoing support to ensure their success in Precalculus.

At Refresh Kid, we believe that a solid foundation in Precalculus is crucial for students' future success in advanced mathematics and other STEM-related fields. Enroll in our tutoring program today and empower your child with the skills and knowledge needed to excel in Precalculus and beyond.

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