Distance and Circles

Distance Between Two Points

The distance between two points in the coordinate plane can be found using the distance formula, which is derived from the Pythagorean Theorem.

• Distance Formula: For points and , the distance is given by:

• Example: Find the distance between and .

• Apply the formula:

Equations of Circles

The standard form of the equation of a circle with center and radius is:

• Center:

• Radius:

• Example: A circle with center and radius has the equation .

Finding Center and Radius from Equation

• Given an equation in the form , the center is and the radius is .

• Example: has center and radius .

Standard Form of a Circle

• To write the equation of a circle in standard form, complete the square if necessary.

• Example: is already in standard form.

Functions and Relations

Definition of a Function

• A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

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

Domain and Range

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

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

• Example: For , the domain is .

Evaluating Functions

• To evaluate a function, substitute the given value for the variable.

• Example: If , then .

Linear Equations and Slope

Slope of a Line

• Slope Formula: For points and :

• Example: The slope between and is .

Slope-Intercept Form

• The slope-intercept form of a line is , where is the slope and is the y-intercept.

• Example: A line with slope passing through : .

Parallel and Perpendicular Lines

• Parallel lines have equal slopes.

• Perpendicular lines have slopes that are negative reciprocals: .

• Example: and are parallel because both have slope .

Solving Equations and Inequalities

Solving Linear Equations

• To solve , isolate on one side.

• Example:

• Expand:

• Rearrange:

Solving and Graphing Inequalities

• To solve inequalities, use similar steps as equations, but reverse the inequality sign when multiplying or dividing by a negative.

• Example:

• Expand:

• Rearrange:

• Graph the solution on a number line, using open or closed circles as appropriate.

Interval Notation

• Interval notation is used to describe sets of numbers between two endpoints.

• Example: means .

Domain and Range from Graphs

• To find the domain and range from a graph, look at the extent of the graph along the -axis (domain) and -axis (range).

• Example: If a graph extends from to , the domain is .

• If the graph extends from to , the range is .

Summary Table: Key Formulas and Concepts

Additional info:

• This review covers foundational Precalculus topics including analytic geometry, functions, linear equations, and inequalities, all of which are essential for further study in mathematics.