Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.
$\text{ƒ}\left(x\right)=x\mid x\mid$

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. A graph is symmetric about the y-axis if replacing x with -x yields the same function value, indicating even symmetry. It is symmetric about the x-axis if replacing y with -y gives the same x value, and it is symmetric about the origin if replacing both x and y with their negatives results in the same function.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is crucial in determining symmetry because it affects the behavior of the graph, particularly in how it reflects across the axes. For example, the function f(x) = x|x| combines linear and absolute value characteristics, influencing its symmetry properties.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. This process helps in identifying symmetry and other characteristics of the graph. By graphing the function f(x) = x|x|, one can observe its shape and confirm its symmetry properties, providing a practical method to validate theoretical findings.