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An absolute value function is a mathematical expression that contains an absolute value, typically represented as |x|. This function outputs the non-negative value of x regardless of whether x is positive or negative. The equation for a basic absolute value function is
$$f(x) = |x|$$
which creates a 'V' shape when graphed. This particular shape is crucial as it directly links to the concept of symmetry in functions. When a function like
$$f(x)=2|x|$$
is used, it affects the 'V' shape by stretching it vertically by a factor of 2, but the core property of absolute value functions—producing non-negative outputs—remains unchanged.

Graphing functions is a visual way of representing the relationship between variables. By plotting a function on a coordinate system, we can visually analyze the behavior of the function, such as intervals of increase and decrease, intercepts, and—as is key in this context—symmetry. Graphing $$f(x)=2|x|$$ involves two separate lines—starting from the origin, one line ascends with a slope of 2, and the other descends with a slope of -2, mirroring each other across the y-axis.
When graphing, consider these steps:

• Identify the type of function and its basic shape.
• Find key points such as intercepts and vertices.
• Plot these points and sketch the function considering its behavior and transformations.

Symmetry about the x-axis occurs when for every point (x,y) on the graph, there is an equivalent point (x,-y). In simple terms, if you can flip the graph over the x-axis and it remains unchanged, the graph is symmetric about the x-axis. However, most functions do not exhibit this type of symmetry. As seen in the exercise solution,
$$f(x) = 2|x|$$
is a rare example where the function exhibits this symmetry due to the absolute value affecting only the magnitude, and not the sign, of the x-values.

Symmetry about the y-axis implies that for every (x,y) on a graph, a corresponding point (-x,y) mirrors it across the y-axis. To check for this symmetry algebraically, replace x with -x in the function's equation and simplify. If the resultant expression is equivalent to the original, the function has y-axis symmetry. The absolute value function
$$f(x)=2|x|$$
shows this symmetry, as negative inputs yield the same output as their positive counterparts—further strengthening that absolute values naturally align with y-axis symmetry.

A graph shows origin symmetry, also known as rotational symmetry, if it looks the same after a rotation of 180 degrees about the origin. To check, you can replace both x and y in the function with their negatives and if the result is the original function with y negated, the function has origin symmetry. However, for the function
$$f(x)=2|x|$$,
such negation will not alter the function, making our intuitive thought of origin symmetry incorrect. This highlights an important nuance: While the function does not change when x is replaced with -x, it does not exhibit true origin symmetry, as flipping both axes does not yield the original equation, due to the dependent value of the absolute function always being positive.