Overview
This lecture provides a comprehensive introduction to College Algebra, covering exponent rules, simplifying expressions, solving equations and inequalities, graphing, quadratic equations, systems of equations, functions, and inverse functions.
Exponents and Simplifying Expressions
- When multiplying like bases, add exponents: (x^2 \times x^5 = x^7).
- When dividing like bases, subtract exponents: (x^5 / x^2 = x^3); (x^4 / x^7 = x^{-3} = 1/x^3).
- Raising a power to a power multiplies the exponents: ((x^3)^4 = x^{12}).
- Any nonzero number raised to the zero power is 1: (x^0 = 1).
Combining Like Terms and Polynomials
- Combine like terms by adding/subtracting their coefficients.
- Distribute negative signs across parentheses before combining terms.
- Monomial: 1 term; Binomial: 2 terms; Trinomial: 3 terms; Polynomial: many terms.
Operations with Polynomials
- Multiply binomials using FOIL (First, Outside, Inside, Last).
- Expand ((a+b)^2) as (a^2 + 2ab + b^2).
Solving Linear Equations and Inequalities
- Use inverse operations to isolate variables: addition/subtraction, multiplication/division.
- For inequalities, reverse the sign when multiplying or dividing by a negative.
- Graph inequalities on a number line: open circle for (<,>), closed circle for (\leq, \geq).
- Interval notation: use parentheses for open, brackets for closed intervals.
Absolute Value Equations and Inequalities
- Absolute value outputs are always non-negative.
- For (|x| = a), (x = a) or (x = -a).
- For inequalities, write two equations to represent both possibilities.
Graphing Linear Equations and Functions
- Slope-intercept form: (y = mx + b) (m = slope, b = y-intercept).
- Standard form: (ax + by = c); find x- and y-intercepts to plot.
- Vertical/horizontal shifts and reflections modify graphs accordingly.
Quadratic Functions and Transformations
- Parabola parent function: (y = x^2), opens upwards.
- Negative signs flip the graph; (y = -x^2) opens downward.
- (y = (x - h)^2 + k) shifts the vertex to ((h, k)).
Solving Quadratic Equations
- Factor using difference of squares or trinomials; set each factor to zero to solve.
- Quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
- Use factoring by grouping for trinomials with leading coefficient (\neq 1).
Complex Numbers
- (i = \sqrt{-1}); (i^2 = -1); (i^3 = -i); (i^4 = 1).
Systems of Equations
- Solve by elimination (add/subtract equations to eliminate a variable).
- Solve by substitution (replace one variable with its expression from the other equation).
- Graphical method: intersection point is the solution.
Functions and Function Operations
- Plug in given (x) values to evaluate the function.
- Synthetic division is an alternate method for evaluating polynomials.
- Composite functions: (f(g(x))) means substitute (g(x)) into (f(x)).
Inverse Functions
- Inverse function swaps x and y values and solves for y.
- Graphically, inverse functions are symmetric over (y = x).
- Two functions are inverses if both (f(g(x)) = x) and (g(f(x)) = x).
Key Terms & Definitions
- Exponent — A number indicating how many times to multiply the base.
- Foil — Method to multiply two binomials.
- Absolute Value — The non-negative value of a number.
- Slope-Intercept Form — Linear equation written as (y = mx + b).
- Quadratic Equation — Equation involving (x^2).
- Quadratic Formula — Formula to solve any quadratic equation.
- System of Equations — Set of equations with multiple variables.
- Function — A rule that assigns one output to each input.
- Inverse Function — Function that "undoes" the effect of the original.
Action Items / Next Steps
- Practice simplifying exponents, combining like terms, and solving linear/quadratic equations.
- Complete assigned homework on graphing and solving inequalities.
- Review key function and inverse function concepts before the next class.