Course Overview: College Algebra (Math 1120)

This study guide summarizes the main topics and subtopics covered in a typical College Algebra course, as outlined in the provided course schedule. Each topic is foundational for understanding algebraic concepts and their applications in mathematics and related fields.

Chapter 1: Functions, Graphs, and Models; Linear Functions

1.1 Functions and Models

Functions are mathematical relationships that assign each input exactly one output. Models use functions to represent real-world phenomena.

• Definition of a Function: A function f from set A to set B is a rule that assigns to each element x in A exactly one element f(x) in B.

• Function Notation: f(x) denotes the output of function f for input x.

• Domain and Range: The domain is the set of all possible inputs; the range is the set of all possible outputs.

• Example: If f(x) = 2x + 3, then f(1) = 5.

1.2 Graphs of Functions

Graphs visually represent the relationship between variables in a function.

• Cartesian Plane: Functions are graphed on the x-y coordinate plane.

• Key Features: Intercepts, slope, symmetry, and intervals of increase/decrease.

• Example: The graph of f(x) = x^2 is a parabola opening upwards.

1.3 Linear Functions

Linear functions are functions of the form f(x) = mx + b, where m is the slope and b is the y-intercept.

• Slope: Measures the steepness of the line.

• Y-intercept: The point where the line crosses the y-axis.

• Equation:

• Example: For f(x) = 2x + 1, the slope is 2 and the y-intercept is 1.

1.4 Equations of Lines

Equations of lines can be written in various forms to describe their properties.

• Slope-Intercept Form:

• Point-Slope Form:

• Standard Form:

• Example: A line passing through (2, 3) with slope 4:

Chapter 2: Linear Models, Equations, and Inequalities

2.1 Algebraic and Graphical Solution of Linear Equations

Linear equations can be solved algebraically or by interpreting their graphs.

• Solving Algebraically: Isolate the variable using inverse operations.

• Graphical Solution: The solution is the x-value where the graph crosses the x-axis.

• Example: Solve ; .

2.2 Fitting Lines to Data Points: Modeling Linear Functions

Linear models are used to fit a line to a set of data points, often using the method of least squares.

• Line of Best Fit: The line that best represents the trend in the data.

• Equation: where m and b are determined from the data.

• Application: Predicting future values based on the model.

2.3 Systems of Linear Equations in Two Variables

Systems of equations involve finding values that satisfy two or more equations simultaneously.

• Methods: Substitution, elimination, and graphical methods.

• Solution Types: One solution (intersecting lines), no solution (parallel lines), infinitely many solutions (same line).

• Example: Solve

2.4 Solutions of Linear Inequalities

Linear inequalities describe a range of possible solutions rather than a single value.

• Solving: Use similar steps as equations, but reverse the inequality when multiplying/dividing by a negative.

• Graphing: Solutions are shown as shaded regions on the number line or coordinate plane.

• Example: Solve ; .

Chapter 3: Quadratic, Piecewise-Defined, and Power Functions

3.1 Quadratic Functions; Parabolas

Quadratic functions are polynomials of degree two and their graphs are parabolas.

• Standard Form:

• Vertex: The highest or lowest point on the parabola.

• Axis of Symmetry:

• Example:

3.2 Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula:

• Factoring: Express as a product of binomials.

• Example: Solve ; solutions are and .

3.3 Power and Root Functions

Power functions have the form ; root functions involve radicals.

• Power Function:

• Root Function:

• Example:

3.4 Piecewise-Defined Functions and Absolute Value Functions

Piecewise-defined functions use different expressions for different intervals of the domain. Absolute value functions measure distance from zero.

• Piecewise Function:

• Absolute Value Function:

• Graph: The graph of is a "V" shape.