Overview
This lecture covers key concepts in Algebra 2, focusing on solving linear equations, inequalities, graphing functions, exponents, quadratic equations, complex numbers, polynomials, rational expressions, and logarithms.
Solving Linear Equations
- Isolate variables by performing inverse operations to both sides (addition/subtraction, multiplication/division).
- For equations with variables on both sides, collect like terms and solve for the variable.
- Use cross-multiplication for equations involving fractions.
Inequalities & Graphing
- Solve inequalities with similar steps as equations, but reverse the inequality sign when multiplying/dividing by a negative.
- Graph solutions on a number line using open/closed circles and shading in the appropriate direction.
- Express solution sets using interval notation, with brackets for inclusion and parentheses for exclusion or infinity.
Absolute Value Concepts
- The absolute value of a number is always non-negative.
- To solve |x| = a, write two equations: x = a and x = -a.
- For absolute value inequalities, set up two inequalities and solve separately, remembering to reverse signs as needed.
Graphing Linear and Absolute Value Functions
- Slope-intercept form: y = mx + b (m = slope, b = y-intercept); plot b, then use m to find other points.
- Standard form: Find x- and y-intercepts for graphing.
- Absolute value graphs are "V" shaped, shifted by transformations based on their equations.
Systems of Equations
- Solve using elimination (add/subtract equations to cancel variables) or substitution (replace variables).
- Solution is the intersection point of the two lines.
Quadratic Equations & Graphs
- Factor quadratics to find roots or use the quadratic formula: x = [-b ± √(b²-4ac)]/(2a).
- The axis of symmetry is x = -b/2a; the vertex is at this x-value.
- Standard, vertex, and factored forms offer different graphing insights.
Exponents and Radicals
- Product rule: x^a * x^b = x^(a+b); quotient: x^a / x^b = x^(a-b); power: (x^a)^b = x^(ab).
- Negative and fractional exponents relate to reciprocals and roots.
- Simplify radicals by factoring out perfect squares and rationalize denominators by multiplying by conjugates.*
Rational Expressions
- Factor numerators/denominators to simplify or multiply/divide expressions.
- For addition/subtraction, find a common denominator and combine numerators.
Complex Numbers
- Standard form: a + bi.
- i^2 = -1; higher powers of i repeat every four exponents.
- Add, subtract, multiply, and divide using arithmetic and FOIL, rationalizing denominators with i.
Polynomials and Finding Zeros
- Factor polynomials using grouping, synthetic division, or quadratic formula for higher degrees.
- The Rational Root Theorem helps list possible rational zeros.
Functions: Domain, Range, Inverse & Composite
- Domain: all possible x-values; range: all possible y-values.
- Inverse functions switch x and y; test by checking if f(g(x)) = x and g(f(x)) = x.
- Composite functions: plug one function into another, f(g(x)).
Logarithms
- log_b(a) = c means b^c = a.
- Properties: log(ab) = log a + log b; log(a/b) = log a - log b; log(a^n) = n log a.
- Change of base: log_b(a) = log_c(a) / log_c(b).
- Solve log equations by converting to exponential form.
Key Terms & Definitions
- Slope-intercept form — y = mx + b, equation of a line.
- Axis of symmetry — The vertical line x = -b/2a for parabolas.
- Conjugate — For a + bi, the conjugate is a - bi.
- Rational Root Theorem — Lists possible rational zeros of polynomials.
- Interval notation — Expresses intervals using brackets/parentheses.
- Absolute value — Distance from zero, always non-negative.
- Composite function — Combining two functions, f(g(x)).
- Inverse function — Function that reverses another function.
Action Items / Next Steps
- Practice solving equations and inequalities of various types.
- Graph different functions (linear, quadratic, absolute value, rational).
- Review factoring, solving quadratics, and manipulating exponents and radicals.
- Complete assigned homework and practice problems to reinforce concepts.