Overview
This lecture covers key concepts of Algebra 2, including linear equations, inequalities, graphing, exponents, quadratic equations, functions, logarithms, radicals, rational expressions, and graphing techniques.
Solving Linear Equations
- To isolate x in linear equations, use inverse operations (add/subtract, multiply/divide).
- Combine like terms and move variables to one side, constants to the other.
- For equations with fractions, multiply both sides by the least common multiple to clear denominators.
- Cross multiplication can be used when both sides are single fractions.
Inequalities & Graphing
- Inequalities use open or closed circles on number lines (open for < or >, closed for ≤ or ≥).
- Interval notation: parentheses for non-inclusive, brackets for inclusive endpoints.
- When multiplying/dividing by a negative, reverse the inequality sign.
Absolute Value Equations & Inequalities
- |x| = a leads to x = a or x = –a.
- Get the absolute value by itself before splitting into two equations.
- For inequalities, create two separate inequalities considering both positive and negative cases.
Graphing Linear & Absolute Value Functions
- Slope-intercept form: y = mx + b, m = slope, b = y-intercept.
- Standard form: Ax + By = C; find x- and y-intercepts for graphing.
- Absolute value graphs are "V" shaped; transformations shift/reflect the graph.
Slope & Linear Equation Forms
- Slope formula: (y₂–y₁)/(x₂–x₁).
- Linear equation forms: slope-intercept (y = mx + b), standard (Ax + By = C), point-slope (y – y₁ = m(x – x₁)).
- Parallel lines have equal slopes; perpendicular slopes are negative reciprocals.
Solving Systems of Equations
- Use elimination, substitution, or graphic intersection to find x and y.
- Align variables for elimination; substitute when one equation is solved for a variable.
Quadratic Equations & Graphing Parabolas
- Factor quadratics (difference of squares, trinomials, grouping).
- Use quadratic formula: x = (–b ± √(b²–4ac))/(2a).
- Complete the square to solve or rewrite standard to vertex form.
- Vertex: axis of symmetry at x = –b/(2a); minimum/maximum at vertex.
- Parabola opens up if leading coefficient > 0, down if < 0.
Function Properties & Inverses
- Domain: all possible x-values; range: all possible y-values.
- Inverse functions: swap x and y, solve for y.
- Two functions are inverses if f(g(x)) = x and g(f(x)) = x.
Exponents & Radicals
- Laws: x^a * x^b = x^(a+b); x^a / x^b = x^(a–b); (x^a)^b = x^(ab).
- Negative exponents: x^(–a) = 1/x^a.
- Fractional exponents: x^(m/n) = n-th root of (x^m).
Logarithms
- log_a(b) = c means a^c = b.
- Properties: log_a(MN) = log_a(M) + log_a(N), log_a(M/N) = log_a(M) – log_a(N), log_a(b^n) = n*log_a(b).
- Change of base: log_a(b) = log_c(b)/log_c(a).
Operations with Complex Numbers
- Standard form: a + bi.
- i = √–1, i² = –1; use FOIL and combine like terms.
- To eliminate complex denominators, multiply numerator and denominator by the conjugate.
Rational & Radical Expressions
- Simplify by factoring numerators/denominators, cancel common factors.
- Rationalize denominators by multiplying by the conjugate or appropriate radical.
- To add/subtract, use the least common denominator.
Solving Radical & Rational Equations
- Isolate the radical, square both sides, solve for x.
- For rational equations, multiply both sides by the common denominator to clear fractions.
Graphing Rational & Radical Functions
- Key features: vertical/horizontal asymptotes, holes, domain/range restrictions.
- For y = 1/(x–h) + k, vertical asymptote at x = h, horizontal at y = k.
Key Terms & Definitions
- Slope — The ratio of vertical change to horizontal change between two points.
- Vertex — The turning point of a parabola.
- Asymptote — A line that a graph approaches but never reaches.
- Domain — Set of all possible input (x) values for a function.
- Range — Set of all possible output (y) values for a function.
- Quadratic Formula — Solution for ax² + bx + c = 0: x = (–b ± √(b²–4ac))/(2a).
- Absolute Value — The distance from zero; always nonnegative.
- Conjugate — For a + bi, the conjugate is a – bi.
Action Items / Next Steps
- Practice solving linear, quadratic, radical, and rational equations.
- Review graphing functions (linear, absolute value, quadratic, rational).
- Practice simplifying exponents, radicals, and working with complex numbers.
- Complete assigned homework problems and review textbook readings on these topics.