Nyack High School

PreCalculus – Curriculum

1. Mathematical Induction - proving statements using mathematical induction
2. System of Equations – Linear
   1. Substitution
   2. Elimination
   3. Identify Inconsistent Systems of Equations Containing 2 Variables
   4. Express the Solution of a System of Dependent Equations Containing 2 Variables
   5. Solve Systems of Three Equations Containing Three Variables
   6. Identify Inconsistent Systems of Equations Containing Three Variables
   7. Express the Solution of a System of Dependent Equations Containing 3 Variables
   8. *Row Echelon Form
3. Systems of Equations & Inequalities
   1. Write the Augmented Matrix of a System of Linear Equations
   2. Write the System of Equations from the Augmented Matrix
   3. Perform Row Operations on a Matrix
   4. Solve a System of Linear Equations Using Matrices
   5. Determinants – 2 by 2, 3 by 3
      1. Evaluate 2 by 2 Determinants
      2. Using Cramer’s Rule to Solve a System of Two Equations Containing 2 Variables

• *Evaluate 3 by 3 Determinants

1. *Use Cramer’s Rule to Solve a System of Three Equations Containing 3 Variables

1. Matrix Algebra
   1. Find the Sum and Difference of Two Matrices
   2. Find Scalar Multiples of a Matrix

• Find the Product of Two Matrices

1. Find the Inverse of a Matrix
2. Solve a System of Linear Equations Using an Inverse Matrix

1. Systems of Nonlinear Equations
   1. Solve a System of Nonlinear Equations Using Substitution
   2. Solve a System of Nonlinear Equations Using Elimination
2. Systems of Inequalities
   1. Graph an Inequality
   2. Graph a System of Inequalities
3. *Linear Programming
   1. Set up a Linear Programming Problem
   2. Solve a Linear Programming Problem
4. Functions
   1. Find the Difference Quotient of a Function
   2. Find the Domain/Range
   3. Operations of Two Functions
   4. Properties of Functions
      1. Determine Even and Odd Functions from a Graphs & Equation
      2. Use a Graph to determine whether a function is Increasing, Decreasing, or Constant

• Use a Graph to Locate Local Maxima and Local Minima

1. Use a Graph to Locate the Absolute Maximum and Absolute Minimum
2. Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine where a function is Increasing or Decreasing
3. Find the Average Rate of Change of a Function

1. Library of Functions – Piecewise-defined Functions & Greatest Integer Function
2. Graphing Techniques – Transformations
   1. Graph Functions Using Vertical and Horizontal Shifts
   2. Graph Functions Using Compressions and Stretches

• Graph Functions Using Reflections about the x-axis and y-axis

5. Polynomial Functions
   1. Polynomial Functions and Models
      1. Identify Polynomial Functions and Their Degree
      2. Graph Polynomial Functions Using Transformations

• Know Properties of the Graph of a Polynomial Function

1. Analyze the Graph of a Polynomial Function
2. Build Cubic Models

1. The Real Zeros of a Polynomial Function
   1. Use the Remainder and Factor Theorems
   2. Use Decartes’ Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function

• Use the Rational Zeros Theorems to List the Potential Rational Zeros of a Polynomial Function

1. Find the Real Zeros of a Polynomial Function
2. Solve Polynomial Equations
3. Use the Theorem for Bounds on Zeros

• Use the Intermediate Value Theorem

1. Complex Zeros; Fundamental Theorem of Algebra
   1. Use the Conjugate Pairs Theorem
   2. Find a Polynomial Function with Specified Zeros

• Find the Complex Zeros of a Polynomial Function

6. Rational
   1. Operations of Rational Functions
   2. Partial Fraction Decomposition
      1. Decompose P/Q where Q has only nonrepeated Linear Factors
      2. Decompose P/Q where Q has repeated Linear Factors
      3. Decompose P/Q where Q has a nonrepeated Irreducible Quadratic Factor
      4. Decompose P/Q where Q has a repeated Irreducible Quadratic Factor
   3. Properties of Rational Numbers
      1. Find the Domain of a Rational Function
      2. Find the Vertical Asymptotes of a Rational Function

• Find the Horizontal or Oblique Asymptote of a Rational Function

1. The Graph of a Rational Function
   1. Analyze the Graph of a Rational Function
   2. Solve Applied Problems Involving Rational Functions
2. Polynomial and Rational Inequalities
   1. Solve Polynomial Inequalities
   2. Solve Rational Inequalities
3. Exponential & Logarithmic Functions
   1. Composite Functions
   2. One-to-One Functions; Inverse Functions
   3. Exponential Functions
   4. Logarithmic Functions
   5. Properties of Logarithms
   6. Logarithmic and Exponential Equations
   7. Financial Models
   8. Exponential Growth and Decay Models; Newton’s Law; *Logistic Growth and Decay Models
   9. Building Expression, *Logarithmic, and *Logistics Models from Data
4. Trigonometric Functions
   1. Angles and Their Measure (review)
   2. Trigonometric Functions; Unit Circle Approach – Special Right Triangle
   3. Properties of the Trigonometric Functions
      1. Determine the Domain and the Range of the Trigonometric Functions
      2. Even and Odd Functions
   4. Graphs of the Sine and Cosine Functions (review)
   5. Graphs of the Tangent, Cotangent, Cosecant, and the Secant Functions
   6. Phase Shift; Sinusoidal Curve Fitting
5. Analytic Trigonometry
   1. The Inverse Sine, Cosine, and Tangent Functions
   2. The Inverse Trigonometric Functions
   3. Trigonometric Equations
   4. Trigonometric Identities
   5. Sum and Difference Formulas
   6. Double-angle and Half angle Formulas
   7. Product-to-Sum and Sum-to-Product Formulas
6. Applications of Trigonometric Functions (Law of Sines/Cosines)
   1. Ambiguous Case
   2. Resultant Forces
7. Analytic Geometry
   1. Conics
   2. The Parabola
      1. Analyze Parabolas with Vertex at the Origin
      2. Analyze Parabolas with Vertex at (h, k)

• Solve Applied Problems Involving Parabolas

1. The Ellipse
   1. Analyze Ellipses with Center at the Origin
   2. Analyze Ellipses with Center at (h, k)

• Solve Applied Problems Involving Ellipses

1. The Hyperbola
   1. Analyze Hyperbolas with Center at the Origin
   2. Analyze Hyperbolas with Center at (h, k)

• Solve Applied Problems Involving Hyperbolas

12. Parametric Equations
    1. Graph Parametric Equations
    2. Find a rectangular Equation for a Curve Defined Parametrically
    3. Use Time as a Parameter in Parametric Equations
    4. Find Parametric Equations for Curves Defined by Rectangular Equations
13. Polar Coordinates; Vectors
    1. Polar Coordinates
       1. Plot Points Using Polar Coordinates
       2. Convert from Polar Coordinates to Rectangular Coordinates

• Convert from Rectangular Coordinates to Polar Coordinates

1. Transform Equations between Polar and Rectangular Forms

1. Polar Equations and Graphs
   1. Identify and Graph Polar Equations by Converting to Rectangular Equations
   2. Test Polar Equations for Symmetry

• Graph Polar Equations by Plotting Points

1. The Complex Plane; De Moivre’s Theorem
   1. Plot points in the Complex Plane
   2. Convert a Complex Number between Rectangular Form and Polar Form

• Find Products and Quotients of Complex Numbers in Polar Form

1. Use DeMoivre’s Theorem
2. Find Complex Roots

1. Vectors
   1. Graph Vectors
   2. Find a Position Vector

• Add and Subtract Vectors Algebraically

1. Find a Scalar Multiple and the Magnitude of a Vector
2. Find a Unit Vector
3. Find a Vector from Its Direction and Magnitude

• Model with Vectors

14. Probability
    1. Real World Applications & Interpretation