Overview
This lecture provides a comprehensive overview of key concepts in College Algebra, including exponents, simplifying expressions, equations, inequalities, graphing, factoring, and functions.
Exponents and Operations
- 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).
- Negative exponents represent reciprocals: (x^4/x^7 = 1/x^3).
- Power of a power: ((x^3)^4 = x^{12}).
- Any nonzero number to the zero power is 1.
Simplifying Expressions & Combining Like Terms
- Combine terms with same variables and exponents.
- Distribute negative signs carefully across parentheses.
- Monomial: 1 term; Binomial: 2 terms; Trinomial: 3 terms; Polynomial: many terms.
Operations with Polynomials
- Use FOIL for multiplying binomials.
- Expand squared binomials by multiplying the expression by itself.
Solving Linear Equations
- Use inverse operations: addition/subtraction and multiplication/division.
- Distribute and combine like terms when needed.
Solving and Graphing Inequalities
- Treat inequalities like equations, but flip the sign when multiplying/dividing by a negative.
- Graph solutions using open (for <, >) or closed (for ≤, ≥) circles.
- Interval notation uses parentheses for open ends, brackets for closed.
Absolute Value Equations & Inequalities
- Absolute value turns negative inputs positive.
- Solve (|x| = a) by setting (x = a) and (x = -a).
- For inequalities, split into two cases and solve each.
Graphing Linear Equations & Transformations
- Slope-intercept form: (y = mx + b); m = slope, b = y-intercept.
- Standard form: use intercepts for plotting.
- Transformations move or reflect parent functions in the plane.
Quadratic Functions and Factoring
- Parent function (y = x^2) forms a parabola.
- Negative leading coefficient reflects parabola downward.
- Factoring difference of squares: (x^2 - a^2 = (x-a)(x+a)).
- Factor trinomials by finding two numbers that multiply to c and sum to b.
- Quadratic formula: (x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}).
Complex Numbers
- (i = \sqrt{-1}); (i^2 = -1); (i^3 = -i); (i^4 = 1).
Systems of Equations
- Elimination: align terms and add/subtract equations to eliminate a variable.
- Substitution: solve for one variable and substitute into the other equation.
- The intersection point represents the solution.
Functions and Function Notation
- Evaluate by plugging in the input value.
- For (f(x)=y), set output equal to y and solve for (x).
Composite and Inverse Functions
- Composite: (f(g(x))) means substitute g(x) into f(x).
- Inverse: Swap x and y, then solve for y.
- Two functions are inverses if (f(g(x)) = g(f(x)) = x).
Key Terms & Definitions
- Exponent — number indicating repeated multiplication of a base.
- Like Terms — terms with identical variable parts.
- FOIL — First, Outer, Inner, Last (order of multiplying binomials).
- Absolute Value — the nonnegative value of a number.
- Slope-Intercept Form — linear equation form (y=mx+b).
- Quadratic Formula — solution for (ax^2 + bx + c=0): (x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}).
- Composite Function — applying one function to the result of another.
- Inverse Function — "undoes" the effect of the original function.
- Complex Number — includes real and imaginary parts.
Action Items / Next Steps
- Practice combining like terms, factoring, and solving quadratic equations.
- Review constructing and graphing linear and quadratic functions.
- Complete assigned problems on systems of equations and function notation.