Algebra 2 Lecture Notes

Solving Linear Equations

• Linear Equation Example:
  • Equation: 5x - 4 = 11
  • Solution: Add 4 to both sides, then divide by 5.
  • Result: x = 3
• Variables on Both Sides Example:
  • Equation: 3x - 7 = 9x + 17
  • Solution: Move x terms to one side, constants to the other.
  • Result: x = -4

Solving Equations with Distribution

• Example: 5 - 2(3x + 1) = 7(2x + 5) - 8
  • Distribute values, combine like terms, solve for x.
  • Result: x = -6/5

Solving Equations with Fractions

• Example: 2/(3x - 5) = 3
  • Combine like terms, multiply by the denominator.
  • Result: x = 12
• Example: 12x - 1/3 = 4
  • Multiply by the common denominator.
  • Result: x = 26/3

Inequalities

• Graphing Inequalities
  • Greater than uses open circle, shade right.
  • Less than uses open circle, shade left.
  • Include or equal to uses closed circle.
• Solving Inequalities
  • Solve and graph solutions, use interval notation for answers.

Absolute Value Equations

• Basic Concept: Absolute value results in positive numbers or zero.
• Solving Absolute Value Equations:
  • Break into two separate equations, solve each.
• Example: |2x - 3| = 6
  • Results: x = 4.5 or x = -1.5

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b
  • Plot b on y-axis, use m to find next points.
• Standard Form: Ax + By = C
  • Find x and y intercepts, plot and connect.

Graphing Inequalities

• Dashed vs Solid Lines:
  • Dashed for < or >, solid for ≤ or ≥.
• Shading:
  • Shade above for >, below for <.

Graphing Absolute Value Functions

• Basic v shape, transformed by shifting, reflecting.

Solving Quadratic Equations

• Factoring Quadratics:
  • Use zero product property: Set each factor equal to zero.
• Difference of Squares:
  • Example: x^2 - 25 = 0 factors to (x + 5)(x - 5).

Quadratic Functions and Graphing

• Vertex Form: y = a(x - h)^2 + k
  • Vertex at (h, k), opens up if a > 0, down if a < 0.
• Finding Vertex:
  • Use x = -b/(2a) to find x-coordinate of vertex.

Polynomials

• Finding Zeros:
  • Use synthetic division or factoring.
• Factoring by Grouping:
  • Useful when ratios of first two terms match last two.

Logarithms

• Basic Concepts:
  • log_b(a) = c translates to b^c = a.
• Properties:
  • Product, Quotient, Power rules.
• Solving Equations:
  • Use properties to simplify and solve.

Rational Expressions

• Simplifying:
  • Factor numerators and denominators, cancel common factors.
• Adding/Subtracting:
  • Find common denominator.

Solving Radical Equations

• Basic Steps:
  • Isolate the radical, square both sides.
• Graphing Radical Functions:
  • Consider shifts and reflections.

Inverse Functions

• Finding Inverse:
  • Switch x and y, solve for new y.

Composite Functions

• Evaluating:
  • Plug one function into another.

Conclusion

• Practice Problems:
  • Encourage solving additional problems for mastery.