Symmetry of Equations and Graphs

Symmetry with Respect to the Axes and Origin

Understanding symmetry helps in graphing and analyzing functions. A graph may be symmetric with respect to the x-axis, y-axis, or the origin.

• Symmetry with Respect to the x-Axis: A graph is symmetric about the x-axis if, for every point (x, y) on the graph, the point (x, -y) is also on the graph.

• Symmetry with Respect to the y-Axis: A graph is symmetric about the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

• Symmetry with Respect to the Origin: A graph is symmetric about the origin if, for every point (x, y) on the graph, the point (-x, -y) is also on the graph.

Testing for Symmetry

To test an equation for symmetry:

• x-axis: Replace y with -y and simplify. If the equation is unchanged, the graph is symmetric about the x-axis.

• y-axis: Replace x with -x and simplify. If the equation is unchanged, the graph is symmetric about the y-axis.

• Origin: Replace x with -x and y with -y and simplify. If the equation is unchanged, the graph is symmetric about the origin.

Example: Test the equation for symmetry.

• x-axis: (symmetric)

• y-axis: (symmetric)

• Origin: (symmetric)

Functions and Their Values

Evaluating Functions

To find the value of a function at a given input, substitute the input value into the function's formula.

• Example: For , find :

The Difference Quotient

Definition and Purpose

The difference quotient of a function is defined as:

• The difference quotient is used in calculus to define the derivative.

Example: For , compute the difference quotient:

Expand and simplify:

So,

Therefore,

Domain of a Function

Finding the Domain

The domain of a function is the set of all real numbers for which the function is defined.

• If the function has a denominator, exclude values that make the denominator zero.

• If the function has an even root (e.g., square root), exclude values that make the radicand negative.

Example: For , the domain is all real numbers except .

Application Example

Given , the domain is (since length cannot be negative).

Operations on Functions

Sum, Difference, Product, and Quotient of Functions

• Sum:

• Difference:

• Product:

• Quotient:

The domain of the combined function is the intersection of the domains of and , and for quotients, exclude values where .

Example: Operations on Functions

Let and .

• Sum:

• Difference:

• Product:

• Quotient: ,

Obtaining Information from Graphs

Reading Function Values and Intercepts

Given a graph, you can determine:

• Function values: Find by locating on the graph and reading the corresponding value.

• Domain and range: The domain is the set of all -values for which the graph exists; the range is the set of all -values.

• Intercepts: The -intercepts are points where the graph crosses the -axis (). The -intercept is where the graph crosses the -axis ().

Example Table: Types of Symmetry

Key Terms and Definitions

• Even Function: A function is even if for all in the domain. Its graph is symmetric about the y-axis.

• Odd Function: A function is odd if for all in the domain. Its graph is symmetric about the origin.

• Neither: If a function is neither even nor odd, it does not have y-axis or origin symmetry.

Summary Table: Function Operations and Domains

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and tables have been logically grouped and formatted for study purposes.