Properties of Functions and Their Graphs

Objective 1: Determining the Intercepts of a Function

Intercepts are key points where a function's graph crosses the axes. Understanding intercepts helps in graphing and analyzing functions.

• Y-intercept: The y-coordinate of the point where the graph crosses or touches the y-axis. Found by evaluating .

• X-intercepts: The x-coordinates of the points where the graph crosses or touches the x-axis. Found by solving .

• Real Zeros: The x-intercepts are also called real zeros of the function.

• Note: A function can have at most one y-intercept, but may have several x-intercepts.

Example: For , the y-intercept is , and the x-intercepts are found by solving , giving and .

Objective 2: Determining the Domain and Range of a Function from its Graph

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). These can be determined visually from the graph.

• Domain: The interval of x-values covered by the graph, typically written as .

• Range: The interval of y-values covered by the graph, typically written as .

Example: If a graph extends from to and from to , then domain is and range is .

Objective 3: Determining Where a Function is Increasing, Decreasing, or Constant

A function's behavior on intervals can be classified as increasing, decreasing, or constant. This is determined by the direction of the graph.

• Increasing: increases as increases on an interval. The graph rises from left to right.

• Decreasing: decreases as increases on an interval. The graph falls from left to right.

• Constant: remains the same as increases. The graph is a horizontal line.

Example: For on , if rises, it is increasing; if it falls, it is decreasing; if it is flat, it is constant.

Objective 4: Determining Relative Maximum and Relative Minimum Values of a Function

Relative extrema are points where a function reaches a local highest or lowest value within an interval.

• Relative Maximum: Occurs at if changes from increasing to decreasing at . The value is .

• Relative Minimum: Occurs at if changes from decreasing to increasing at . The value is .

• Note: Relative extrema must occur in the interior of the domain, not at endpoints.

Example: For , the relative minimum is at , .

Objective 5: Determining if a Function is Even, Odd, or Neither

Functions can be classified based on their symmetry properties.

• Even Function: for all in the domain. The graph is symmetric about the y-axis.

• Odd Function: for all in the domain. The graph is symmetric about the origin.

• Neither: If neither condition holds, the function is neither even nor odd.

Example: is even; is odd; is neither.

Objective 6: Determining Information about a Function from a Graph

Graphs provide visual information about a function's domain, range, intercepts, intervals of increase/decrease, and symmetry.

• Identify intercepts, extrema, and intervals of increase/decrease.

• Check for symmetry to determine if the function is even or odd.

• Use the graph to estimate values and behavior of the function.

Relations and Functions

Objective 1: Understanding the Definitions of Relations and Functions

Relations and functions are foundational concepts in mathematics, describing how elements from one set are paired with elements from another.

• Relation: A correspondence between two sets, A and B, where each element of A is paired with one or more elements of B.

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

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

• Function: A special type of relation where each element in the domain is paired with exactly one element in the range.

Example: is a function because each has one .

Objective 2: Determine if Equations Represent Functions

To determine if an equation represents a function, check that each input value corresponds to exactly one output value.

• For every in the domain, there must be only one .

• Equations like do not represent functions because some values correspond to two values.

Objective 3: Using Function Notation; Evaluating Functions

Function notation expresses the output of a function for a given input. Evaluating functions means finding the output for a specific input.

• Notation: means the value of function at .

• Difference Quotient: is important in calculus for finding rates of change.

Example: If , then .

Objective 4: Using the Vertical Line Test

The vertical line test is a graphical method to determine if a graph represents a function.

• If any vertical line intersects the graph more than once, it is not a function.

• If every vertical line intersects the graph at most once, it is a function.

Objective 5: Determining the Domain of a Function Given the Equation

The domain of a function is the set of all input values for which the function is defined. Different types of functions have different domain restrictions.

• Polynomial Function: ; domain is all real numbers.

• Rational Function: ; domain is all real numbers except where .

• Root Function: ; domain depends on and :

  • If is even, domain is where .

  • If is odd, domain is all real numbers for which is defined.

Example: For , domain is all real numbers except .

Parallel and Perpendicular Lines

Objective 1: Understanding the Definition of Parallel Lines

Parallel lines in the Cartesian plane do not intersect and have the same slope.

• Theorem: Two distinct non-vertical lines are parallel if and only if they have the same slope.

• If two lines have the same slope, they are parallel.

Example: Lines and are parallel.

Objective 2: Understanding the Definition of Perpendicular Lines

Perpendicular lines intersect at a right angle (90°) and have slopes that are negative reciprocals.

• Theorem: Two non-vertical lines are perpendicular if and only if the product of their slopes is .

• If and are slopes, then .

Example: Lines and are perpendicular.

Objective 3: Determine if Two Lines are Parallel, Perpendicular, or Neither

Compare the slopes of two lines to classify their relationship.

• If slopes are equal, lines are parallel.

• If slopes are negative reciprocals, lines are perpendicular.

• Otherwise, lines are neither parallel nor perpendicular.

Objective 4: Finding the Equations of Parallel and Perpendicular Lines

Given a line and a point, you can find the equation of a line parallel or perpendicular to the given line passing through the point.

• Parallel: Use the same slope as the given line.

• Perpendicular: Use the negative reciprocal of the given line's slope.

• Apply the point-slope form: .

Example: Find the equation of the line parallel to passing through : .