Overview
This lecture reviews key algebra topics, including exponent rules, radicals, factoring, solving equations and inequalities, functions (including inverses), and graphs of common functions.
Exponent and Radical Rules
- An exponent indicates how many times the base is multiplied by itself; the base is the bottom number, the exponent (or power) is the top number.
- Product Rule: xⁿ · xᵐ = xⁿ⁺ᵐ — add exponents when multiplying like bases.
- Quotient Rule: xⁿ / xᵐ = xⁿ⁻ᵐ — subtract exponents when dividing like bases.
- Power Rule: (xⁿ)ᵐ = xⁿ·ᵐ — multiply exponents when raising a power to a power.
- Zero Exponent: x⁰ = 1 (x ≠ 0).
- Negative Exponent: x⁻ⁿ = 1 / xⁿ.
- Fractional Exponent: x¹⁄ⁿ = ⁿ√x (the n-th root of x).
- Exponents distribute over multiplication and division, not addition or subtraction.
- Radical Rules: √ab = √a·√b, √(a/b) = √a/√b, but √(a + b) ≠ √a + √b.
Factoring Techniques
- Greatest Common Factor (GCF): Factor out the largest piece in each term first.
- Factoring by Grouping: Useful for four-term polynomials; group and factor common terms.
- Quadratic Factoring: ax² + bx + c; find two numbers that multiply to ac and sum to b, then factor by grouping.
- Special Formulas: Difference of squares: a² – b² = (a + b)(a – b); Sum/difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²).
- Always factor completely, sometimes using multiple techniques.
Rational Expressions & Equations
- Simplify by factoring numerators/denominators and cancelling common factors.
- Multiply/divide by multiplying numerators and denominators (flip divisor for division).
- Add/subtract by finding a common denominator, usually the least common denominator (LCD).
- For rational equations, clear denominators by multiplying both sides by the LCD, then solve and check solutions (exclude values making denominator zero).
Solving Equations and Inequalities
- For quadratic equations: Write in standard form, factor or use the quadratic formula x = [–b ± √(b²–4ac)]/2a.
- Radical equations: Isolate the radical, raise both sides to a power, solve, and check for extraneous solutions.
- Absolute value equations: Isolate | |, then solve for both positive and negative cases.
- Linear inequalities: Solve like equations, reverse inequality if multiplying/dividing by a negative.
- Interval notation: Use brackets [ ] for inclusive and parentheses ( ) for exclusive; infinity always uses ( ).
Graphs and Functions
- Function: Assigns each input to exactly one output; f(x) used for notation.
- Domain: Set of allowable x-values; exclude division by zero and even roots of negatives.
- Range: Set of output (y) values.
- Toolkit Functions: Memorize basic graphs for linear, quadratic, cubic, square root, exponential, logarithmic, rational, and absolute value functions.
- Transformations: Shifts (inside for x, outside for y), stretches/shrinks (multiplying), reflections (negatives).
- Quadratics: Vertex form y = a(x–h)² + k has vertex (h, k); vertex in standard form is at x = –b/(2a).
- Polynomials: Degree determines end behavior; max turning points = degree – 1.
- Rational functions: Vertical asymptotes where denominator is zero; horizontal asymptotes depend on degree of numerator vs denominator; holes where factors cancel.
Logs and Exponentials
- logₐb = c means aᶜ = b.
- logₐ1 = 0; logₐa = 1.
- Product rule: logₐ(xy) = logₐx + logₐy.
- Quotient rule: logₐ(x/y) = logₐx – logₐy.
- Power rule: logₐ(xⁿ) = n·logₐx.
- Inverse property: logₐ(aˣ) = x, a^(logₐx) = x.
- Exponential equations: Take log to solve for exponents.
- Solve log equations by exponentiating both sides.
Functions: Operations, Composition, and Inverses
- f+g, f–g, f·g, f/g defined by applying the operation to function values at x.
- Composition: (f∘g)(x) = f(g(x)); order matters.
- Inverse function f⁻¹(x): undoes f(x); f(f⁻¹(x)) = x.
- To find inverse: swap x and y, solve for y.
- A function has an inverse only if it passes the horizontal line test (one-to-one).
Action Items / Next Steps
- Practice simplifying exponents and radicals, factoring, and solving equations.
- Graph basic toolkit functions; memorize their shapes.
- Complete assigned problems in the course software (e.g., ALEKS).
- Review log and exponential rules; ensure calculator proficiency.
- Prepare for quizzes on inequalities, functions, and graph transformations.