Study Guide: Algebra 2 Final Exam Preparation

Key Concepts & Problem Solving Techniques

Solving Inequalities

• Problem Example: Solve for x in an inequality.
• Techniques Used:
  • Use the distributive property to simplify expressions.
  • Solve the equation step-by-step, isolating x.
  • Remember when dividing or multiplying by a negative number to flip the inequality sign.
  • Verify solution by testing values.

Systems of Equations

• Problem Example: Solve for x and y using two equations.
• Techniques Used:
  • Substitution Method: Solve one equation for one variable and substitute into the other equation.
  • Elimination Method: Add or subtract equations to eliminate a variable.
  • Verify solution by substituting back into original equations.

Graphing Absolute Value Functions

• Problem Example: Graph y = |x + 3|.
• Concepts Covered:
  • Parent Graphs: Start with the basic graph y = |x| (V-shape).
  • Transformations: Understand how to shift graphs horizontally and vertically.
  • Graphing Technique: Shift the V-graph left by three units.

Simplifying Complex Numbers

• Problem Example: Simplify expressions involving i.
• Techniques Used:
  • Understand i as the square root of -1.
  • Use FOIL method to multiply binomials.
  • Simplify expressions considering i^2 = -1.

Quadratic Equations

• Problem Example: Solve using the quadratic formula.
• Formula: x = (\frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
• Techniques:
  • Identify values for a, b, and c.
  • Plug into the formula and simplify.
  • Interpret plus/minus in solutions to find two possible x values.

Domain and Range

• Concepts: Domain

  • All possible x-values for a function or graph.
  • Typically all real numbers unless restricted by the function.

• Concepts: Range

  • All possible y-values for a function or graph.
  • Find highest and lowest points the graph reaches.

Polynomial Division

• Problem Example: Divide polynomials using long division.
• Steps:
  • Follow steps similar to numerical division: divide, multiply, subtract, bring down.
  • Understand remainder as fractional part of the division.

Function Composition

• Problem Example: Find f(g(x)) for given functions f(x) and g(x).
• Steps:
  • Replace x in f(x) with g(x).
  • Simplify expression using basic algebraic techniques like distribution.

Solving Equations with Radical Expressions

• Problem Example: Solve an equation with a cube root.
• Techniques:
  • Isolate the radical expression.
  • Eliminate radicals by raising both sides to the power of the root.
• Solve resulting equation.

Logarithms

• Concepts: Understanding logs as the inverse of exponents.
• Problem Example: Solve for x given a logarithmic equation.
• Techniques:
  • Trial and error to find exponents manually.
  • Use calculator: Log of the number over log of the base.

Additional Resources

• Videos available for deeper dives into each topic.
• Midterm exam videos for additional practice problems.

Final Notes

• Practice different types of problems for a well-rounded understanding.
• Contact instructor or use additional resources if you have questions.