Math 107 Quiz 31: Is The Graph Symmetric?
Math 107 Quiz 31 6 Pts For Each Graph Is The Graph Symmetric With
For each graph, determine whether it is symmetric with respect to the x-axis, y-axis, and the origin. Provide a simple Yes or No answer for each symmetry type without explanation.
Paper For Above instruction
Symmetry analysis in graphing is a fundamental aspect of understanding functions and their visual representations. Determining whether a graph exhibits symmetry with respect to the x-axis, y-axis, or origin allows mathematicians and students to infer properties about the function without performing extensive calculations. This analysis not only helps in sketching graphs efficiently but also provides insight into the underlying algebraic structure of the functions involved.
Symmetry with respect to the x-axis occurs if replacing y with -y in the function's equation results in an equivalent equation. Mathematically, this implies that if the original function is y = f(x), then the graph is symmetric about the x-axis if for every (x, y) on the graph, (x, -y) is also on the graph. Graphically, this means the top and bottom halves of the graph are mirror images of each other across the x-axis.
Symmetry with respect to the y-axis is observed if replacing x with -x yields the same function value, meaning the graph is a mirror image across the y-axis. Formally, if y = f(x), then the graph is symmetric about the y-axis if for every (x, y) there exists (-x, y) on the graph. This typically indicates that the function is an even function, characterized algebraically by the property f(-x) = f(x).
Symmetry with respect to the origin involves replacing both x and y with their negatives. For a graph to be symmetric about the origin, the condition is that for every (x, y), the point (-x, -y) also lies on the graph. Geometrically, this results in the graph looking the same when rotated 180 degrees about the origin, which corresponds to the property of odd functions where f(-x) = -f(x).
Applying these symmetry tests to specific graphs allows quick classification of the functions. If a given graph meets the symmetry criteria, it can be identified as even, odd, both, or neither. No explanation beyond a simple Yes or No is required, based on the symmetry tests directly inferred from the graph's appearance or the algebraic properties of the function.
References
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