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.
$ƒ (x) = x^{5} - x^{3} - 2$

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Identify the function: $ f(x) = x^5 - x^3 - 2 $.

Check for symmetry about the x-axis by replacing $ f(x) $ with $ -f(x) $ and see if the equation remains unchanged. Since $ f(x) $ is not equal to $ -f(x) $, it is not symmetric about the x-axis.

Check for symmetry about the y-axis by replacing $ x $ with $ -x $ in the function and see if $ f(-x) = f(x) $. Calculate $ f(-x) = (-x)^5 - (-x)^3 - 2 = -x^5 + x^3 - 2 $. Since $ f(-x) \neq f(x) $, it is not symmetric about the y-axis.

Check for symmetry about the origin by replacing $ x $ with $ -x $ and $ f(x) $ with $ -f(x) $. Calculate $ -f(x) = -(x^5 - x^3 - 2) = -x^5 + x^3 + 2 $. Since $ f(-x) = -f(x) $, the function is symmetric about the origin.

Graph the function to visually confirm the symmetry about the origin.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. A graph is symmetric about the x-axis if replacing y with -y yields the same equation, about the y-axis if replacing x with -x does, and about the origin if replacing both x and y with their negatives results in the same equation. Understanding these symmetries helps in analyzing the behavior of functions and their graphs.

Odd and Even Functions

Odd and even functions are classifications based on symmetry. A function is even if f(-x) = f(x), indicating symmetry about the y-axis, while it is odd if f(-x) = -f(x), indicating symmetry about the origin. Recognizing whether a function is odd or even can simplify the process of graphing and analyzing its properties, particularly in calculus.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between the input (x) and output (f(x)). This process helps in identifying key features such as intercepts, slopes, and symmetries. Using graphing tools or software can enhance understanding and provide a clearer picture of the function's behavior, especially when determining symmetries.

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