1. Introduction to Graphs and Coordinates

1.1 Ordered Pairs and the Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points on this plane are represented by ordered pairs (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

• Ordered Pair: A pair of numbers (x, y) that locates a point on the coordinate plane.

• Graph of an Equation: The set of all points (x, y) that satisfy 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).

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

2. Basics of Functions and Their Graphs

2.1 Definition of a Function

A function is a rule that assigns to each input value (usually x) exactly one output value (usually y). The graph of a function is the set of all ordered pairs (x, f(x)).

• Domain: The set of all possible input values (x) for the function.

• Range: The set of all possible output values (f(x)).

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

• Example: For , the domain is all real numbers, and the range is .

2.2 Graphs of Functions

The graph of a function visually represents the relationship between the input and output values. It helps determine the domain, range, and other properties.

• Intercepts: Points where the graph crosses the axes.

• Example: The graph of crosses the y-axis at (0, 0).

3. Equations of Lines and Circles

3.1 Slope and Equations of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x between two points.

• Slope Formula:

• Point-Slope Formula:

• Slope-Intercept Formula:

• Example: The line passing through (1, 2) with slope 3 has equation .

3.2 Distance Formula

The distance formula calculates the distance between two points and in the coordinate plane.

• Example: The distance between (1, 2) and (4, 6) is .

3.3 Equation of a Circle

The equation of a circle with center and radius is:

• Example: A circle centered at (2, -1) with radius 3 has equation .

4. Linear Functions and Their Graphs

4.1 Linear Functions

A linear function is a function of the form , where is the slope and is the y-intercept. Its graph is a straight line.

• Parallel Lines: Have equal slopes.

• Perpendicular Lines: Have slopes that are negative reciprocals.

• Example: Lines with slopes 2 and -1/2 are perpendicular.

5. Transformations of Functions

5.1 Types of Transformations

Transformations change the position or shape of a graph. Common transformations include vertical and horizontal shifts, reflections, and stretching/compressing.

• Vertical Shift: shifts the graph up by units.

• Horizontal Shift: shifts the graph right by units.

• Reflection: reflects the graph across the x-axis; reflects across the y-axis.

• Example: The graph of is the graph of shifted right 2 units and up 3 units.

6. Combinations and Inverse Functions

6.1 Combinations of Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Sum:

• Difference:

• Product:

• Quotient:

• Composition:

• Example: If and , then .

6.2 Inverse Functions

The inverse function of , denoted , reverses the effect of . If , then .

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

• Example: If , then .

7. Summary Table: Key Formulas

Additional info: Students do not need to memorize the basic graphs at the beginning of section 1.6 (e.g., , , etc.), but should understand how transformations affect these graphs.