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Alegbra 2B
   

Course Description

This course is designed to complement and expand upon the mathematical content and concepts of Algebra 2A and Geometry. Students will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic complex number system, matrices and determinants, conic sections and trigonometry. Students are required to master the essential components of Algebra 2B, and to apply these components to hypothetical everyday situations.

Credits: 5


University of California
A-G Approval
State Standards
High School Exit Exam

Course Content

Unit 6 – Series and Sequences

California Mathematics Curriculum Standards

Evaluate Rational Expressions 7.0 – Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.

Graph Quadratic Functions 10.0 – Students graph quadratic functions and determine the maxima, minima, and zeroes of the function.

Understand Exponential Functions 12.0 – Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

Compute Combinations and Permutations 18.0 –Students use fundamental counting principles to compute combinations and permutations.

Compute Probabilities 19.0 –Students use combinations and permutations to compute probabilities.

Expand Binomial Expressions 20.0 – Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.

Apply Induction 21.0 – Students apply the method of mathematical induction to prove general statements about the positive integers.

Find Sums of Series 22.0 –Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

Derive Formulas for Series 23.0 –Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

Unit Objectives
Upon completion of this Unit, students will be able to:

  • Use vocabulary and symbolic notations for sequences and series.
  • Derive sequence formulas and compute missing terms based upon patterns.
  • Use permutations, combinations, and probability to solve problems relating to sequences and series.
  • Solve real world applications relating to sequences and series.
  • Explore patterns using Pascal’s Triangle.
  • Express their understanding through writing exercises.
  • Demonstrate higher level learning through evaluation and presentations of essays and activities.

 

Unit 7 – Exponential and Logarithmic Functions

California Mathematics Curriculum Standards

Evaluate Rational Expressions 7.0 – Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.

Understand Exponents and Logarithms 11.1 – Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Judge Validity 11.2 —Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.

Understand Exponential Functions 12.0 – Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

Translate Between Logarithms 13.0 – Students use the definition of logarithms to translate between logarithms in any base.

Simplify Logarithms 14.0 – Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.

Determine Truth of Algebraic Statements 15.0 – Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.

Unit Objectives
Upon completion of this Unit, students will be able to:

  • Use vocabulary and symbolic notations for exponential and logarithmic functions.
  • Evaluate natural base e and base 10 problems.
  • Sketch graphs of exponential and logarithmic functions.
  • Use and apply properties of logarithms.
  • Solve logarithmic problems by changing bases.
  • Real world applications relating to exponential and logarithmic functions.
  • Express their understanding through writing exercises.
  • Demonstrate higher level learning through evaluation and presentations of reports.

 

Unit 8 – Matrices and Determinants

California Mathematics Curriculum Standards

Solving Systems of Linear Equations and Inequalities 2.0 – Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

Justifying Functions 25.0 – Students use properties from number systems to justify steps in combining and simplifying functions

Unit Objectives
Upon completion of this Unit, students will be able to:

  • Use vocabulary and symbolic notations for matrices.
  • Perform matrix operations.
  • Evaluate determinants.
  • Solve equations and systems of equations using Cramer’s Rule and inverse matrices.
  • Use matrices to solve real world applications.
  • Express their understanding through writing exercises.
  • Demonstrate higher level learning through evaluation and presentations of reports.

 

Unit 9 – Conic Sections

California Mathematics Curriculum Standards

Graph Parabolas 9.0 – Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, student can determine how the graph of a parabola change as a, b, and c vary in the equation .

Graph Conic Sections 16.0 – Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.

Recognize Conic Sections 17.0 -- Students use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola.  Students can then graph the equation.

Unit Objectives
Upon completion of this Unit, students will be able to:

  • Use vocabulary for conic sections.
  • Sketch graphs and determine formulas for conic sections.
  • Write standard equations and classify conic sections.
  • Find equations of conic sections.
  • Use conic sections to solve real world applications.
  • Express their understanding through writing exercises.

 

Unit 10 – Trigonometry

California Mathematics Curriculum Standards

Operations on Rational Expressions 7.0 -- Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.

Operations on Functions 24.0 -- Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.

Combine and Simplify Functions 25.0 -- Students use properties from number systems to justify steps in combining and simplifying functions.

Unit Objectives
Upon completion of this Unit, students will be able to:

    • Use vocabulary for trigonometry.
    • Demonstrate mathematical skills relating to the unit circle and radian measures.
    • Graph and interpret circular and inverse trigonometric functions.
    • Compute values of circular and inverse trigonometric functions.
    • Recognize and prove the basic trigonometric identity cos2 x + sin2 x = 1.
    • Use Laws of Sines and Cosines.
    • Use trigonometry and the geometry of right triangles to find unknown measures and areas.
    • Use trigonometry to solve real world applications.
    • Express their understanding through writing exercises.