Chapter 1: Graphs, Functions, and Models

Introduction to Graphing

Graphing is a foundational skill in algebra, allowing us to visualize equations, functions, and data. The Cartesian coordinate system is used to plot points, graph equations, and analyze relationships between variables.

• Cartesian Coordinate System: Consists of two perpendicular axes (x-axis and y-axis) intersecting at the origin (0, 0). The plane is divided into four quadrants.

• Plotting Points: An ordered pair (x, y) locates a point by moving x units left/right and y units up/down from the origin.

• Solutions of Equations: An ordered pair (x, y) is a solution to an equation in two variables if substituting x and y yields a true statement.

• Graph of an Equation: The set of all solutions plotted in the coordinate plane forms the graph of the equation.

• x-Intercept: The point where the graph crosses the x-axis (y = 0).

• y-Intercept: The point where the graph crosses the y-axis (x = 0).

• Graphing Nonlinear Equations: For equations not in the form Ax + By = C, such as quadratics, plot several points and connect them smoothly.

• Distance Formula: The distance between points (x₁, y₁) and (x₂, y₂) is given by:

• Midpoint Formula: The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is:

• Equation of a Circle: A circle with center (h, k) and radius r has the equation:

Solving Linear Inequalities

Linear inequalities are statements involving <, >, ≤, or ≥. The solution set consists of all values making the inequality true.

• Addition Principle: Adding the same value to both sides preserves the inequality.

• Multiplication Principle: Multiplying both sides by a positive number preserves the inequality; multiplying by a negative reverses the sign.

• Compound Inequalities: Combine two inequalities with 'and' (conjunction) or 'or' (disjunction).

Chapter 2: More on Functions

Increasing, Decreasing, and Constant Functions

Functions can be classified by their behavior over intervals:

• Increasing: for in interval I.

• Decreasing: for in interval I.

• Constant: for all in interval I.

• Relative Maximum/Minimum: A function value is a relative maximum if it is greater than all nearby values; a relative minimum if it is less than all nearby values.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Evaluate by determining which interval the input belongs to and using the corresponding formula.

Greatest Integer Function

The greatest integer function, , returns the largest integer less than or equal to x. Its graph is a step function.

Functions and Graphs

Definition of a Function

A function is a correspondence between a domain and a range such that each input has exactly one output.

• Relation: Any correspondence between two sets; not all relations are functions.

• Function Notation: denotes the output for input x.

Graphing Functions

To graph a function, plot points (x, f(x)) and connect them smoothly. The vertical-line test determines if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

Domain and Range

The domain is the set of all possible inputs (x-values); the range is the set of all possible outputs (y-values). For rational and radical functions, exclude values that make the denominator zero or the radicand negative.

Linear Functions, Slope, and Applications

Linear Functions

A linear function has the form , where m is the slope and b is the y-intercept.

• Slope: measures the steepness of a line.

• Horizontal Lines: (slope 0).

• Vertical Lines: (not a function; slope undefined).

Equations of Lines

• Slope-Intercept Form:

• Point-Slope Form:

• Parallel Lines: Same slope, different y-intercepts.

• Perpendicular Lines: Slopes are negative reciprocals.

Algebra of Functions

Operations with Functions

• Sum:

• Difference:

• Product:

• Quotient: ,

Difference Quotient

The difference quotient is used to compute the average rate of change of a function:

Composition and Decomposition of Functions

Composition of Functions

The composition is defined as . The domain consists of all x in the domain of g such that is in the domain of f.

Decomposition:

Expressing a function as a composition of two or more simpler functions.

Symmetry, Even and Odd Functions

Symmetry

• x-axis: Replace y with –y; if equivalent, symmetric about x-axis.

• y-axis: Replace x with –x; if equivalent, symmetric about y-axis.

• Origin: Replace x with –x and y with –y; if equivalent, symmetric about the origin.

Even and Odd Functions

• Even: for all x in the domain (symmetric about y-axis).

• Odd: for all x in the domain (symmetric about the origin).

Transformations of Functions

Translations

• Vertical: shifts up by b; shifts down by b.

• Horizontal: shifts right by d; shifts left by d.

Reflections

• Across x-axis:

• Across y-axis:

Stretching and Shrinking

• Vertical Stretch: ,

• Vertical Shrink:

• Horizontal Stretch/Shrink: , shrinks, stretches

Variation

Direct Variation

y varies directly as x if for some constant k.

Inverse Variation

y varies inversely as x if for some constant k.

Combined and Joint Variation

• Joint Variation:

• Combined Variation:

Additional info:

Some sections reference applications, modeling, and problem-solving strategies, which are essential for interpreting and applying algebraic concepts in real-world contexts.