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 where the graph crosses or touches the y-axis. Found by evaluating .

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

• Real Zeros: X-intercepts are also called real zeros. Only real solutions to are considered x-intercepts.

• Example: For , the y-intercept is , and the x-intercepts are and .

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

The domain and range describe the set of possible input and output values for a function, respectively.

• Domain: The set of all -values for which the function is defined. On a graph, this is the interval along the -axis covered by the graph.

• Range: The set of all -values that the function attains. On a graph, this is the interval along the -axis covered by the graph.

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

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

Functions can behave differently over various intervals. Identifying these behaviors is essential for graph analysis.

• 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 unchanged as increases. The graph is a horizontal line.

• Example: For on , may be decreasing; on , constant; and on , increasing.

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 open intervals, not at endpoints.

• Example: For , the relative minimum is at , .

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

Functions can exhibit symmetry, which is useful for graphing and analysis.

• 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 behavior, including intercepts, domain, range, and symmetry.

• Key Features: Intercepts, intervals of increase/decrease, relative extrema, and symmetry can all be identified from a graph.

• Example: By examining a graph, one can estimate where is positive, negative, or zero.

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 correspond to elements in 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 is a function, check that each input has exactly one output.

• Vertical Line Test: If any vertical line crosses the graph more than once, it is not a function.

• Example: passes the test; (a circle) does not.

Objective 3: Using Function Notation; Evaluating Functions

Function notation expresses the output of a function for a given input.

• Notation: means the value of function at .

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

• Example: For , .

Objective 4: Using the Vertical Line Test

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

• Test: If any vertical line intersects the graph more than once, the graph does not represent a function.

• Example: The graph of passes; the graph of a circle fails.

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.

• Polynomial Function: ; domain is all real numbers.

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

• Root Function: ; domain depends on :

  • 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 never intersect and have the same slope.

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

• Key Points:

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

  • If two non-vertical 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 .

• Key Points:

  • If the slope of one line is , the slope of a perpendicular line is .

• Example: Lines and are perpendicular.

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

By comparing slopes, one can classify the relationship between two lines.

• Parallel: Same slope.

• Perpendicular: Slopes are negative reciprocals.

• Neither: Any other case.

• Example: and are perpendicular.

Objective 4: Finding the Equations of Parallel and Perpendicular Lines

Given a point and a line, one can write equations for lines parallel or perpendicular to the given line.

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

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

• Example: Find the equation of the line through parallel to :

  • Slope is $3y - 2 = 3(x - 1)$.