1. Introduction to Functions and Their Graphs

1.1 Ordered Pairs and the Coordinate Plane

An ordered pair is a pair of numbers written as (x, y), where the first number is the x-coordinate and the second is the y-coordinate. Ordered pairs are used to locate points on the coordinate plane.

• Plotting Points: To plot (x, y), move x units along the horizontal axis and y units along the vertical axis.

• Example: The point (3, -2) is 3 units right and 2 units down from the origin.

1.2 The Equation of a Graph

The graph of an equation in two variables is the set of all points (x, y) that satisfy the equation. Each solution corresponds to a point on the graph.

• 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).

• Example: For the equation , the y-intercept is (0, 1).

2. Basics of Functions and Their Graphs

2.1 Definition of a Function

A function is a rule that assigns to each element x in a set called the domain exactly one element y in a set called the range. The function is often written as .

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Example: is a function; is not a function.

2.2 Graphs of Functions

The graph of a function is the set of all ordered pairs (x, f(x)).

• Intercepts: Points where the graph crosses the axes.

• Domain and Range from Graph: The domain is the set of all x-values on the graph; the range is the set of all y-values.

3. Equations of Lines and Circles

3.1 Slope of a Line

The slope (m) of a line measures its steepness and is calculated as:

• Interpretation: The change in y divided by the change in x between two points and .

3.2 Point-Slope and Slope-Intercept Forms

• Point-Slope Form: Used to find the equation of a line given a point and the slope.

• Slope-Intercept Form: Used to find y-values or graph the line; b is the y-intercept.

3.3 Distance Formula

The distance between two points and is:

3.4 Equation of a Circle

The equation of a circle with center and radius is:

4. More on Slopes

• Parallel Lines: Have equal slopes.

• Perpendicular Lines: Have slopes that are negative reciprocals.

• Example: If one line has slope 2, a perpendicular line has slope .

5. Transformations of Functions

Transformations change the position or shape of a graph. Common transformations include:

• Vertical Shifts: shifts the graph up by c units.

• Horizontal Shifts: shifts the graph right by c units.

• Reflections: reflects the graph over the x-axis; reflects over the y-axis.

• Vertical Stretch/Compression: stretches (if ) or compresses (if ) the graph vertically.

6. Combinations and Inverse Functions

6.1 Combinations of Functions

Functions can be combined using addition, subtraction, multiplication, and division:

• ,

6.2 Composite Functions

The composition of functions is written as .

• Example: If and , then .

6.3 Inverse Functions

The inverse of a function is denoted and satisfies and for all x in the domain of .

• Finding the Inverse: Solve for x in terms of y, then interchange x and y.

• Example: If , then , so .

7. Summary Table: Forms and Properties of Lines

Additional info: You do NOT need to memorize the basic graphs at the beginning of section 1.6 (e.g., , , etc.).