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Function symmetry is an important concept when graphing rational functions like \( y = -\frac{1}{x^2} \). Symmetry makes it easier to understand how the graph behaves around the axes. For a function to be symmetric with respect to the y-axis, it must satisfy the condition \( f(-x) = f(x) \). This means that the graph on one side of the y-axis is a mirror image of the graph on the other side.
In this case, substituting \(-x\) into the function results in \( y = -\frac{1}{(-x)^2} = -\frac{1}{x^2} \). We observe that the expression remains unchanged, indicating y-axis symmetry. This type of symmetry shows that the behavior of the function at positive x values is mirrored at the negative x values. This insight simplifies the graphing process, offering a better visual understanding of how the function transforms over its domain.

Identifying intervals where the function is increasing or decreasing provides a deeper insight into its behavior. For the function \( y = -\frac{1}{x^2} \), we find it is undefined at \( x = 0 \), but we can analyze the intervals \((-fty, 0)\) and \((0, fty)\).
The function decreases as the absolute value of \( x \) rises, either positively or negatively. This is because as \( |x| \) increases, the magnitude of \( -\frac{1}{x^2} \) becomes larger and thus more negative. Therefore, we say:

• The function decreases over the interval \((-fty, 0)\)
• The function also decreases over \((0, fty)\)

Thus, the entire function is decreasing in both intervals separated by the vertical asymptote at \( x = 0 \). This knowledge is useful for predicting how the graph dives toward negative infinity as \( x \) moves away from zero.

Derivatives play a crucial role in determining where a function is increasing or decreasing. By analyzing the derivative, we gain insight into the function's slope at various points in its domain.
For \( y = -\frac{1}{x^2} \), the derivative is calculated as \( y' = \frac{2}{x^3} \). This derivative is positive when \( x < 0 \) and negative when \( x > 0 \). But note that being positive or negative here refers to the position of the graph rather than increase or decrease directions — which is an essential distinction.
As the derivative here shifts signs across \( x = 0 \), the function itself is entirely decreasing, meaning its downward slope becomes steeper as we move away from zero. From a derivative perspective, we conclude:

• The negative slope of the function suggests a decreasing behavior
• The change in sign at \( x = 0 \) accentuates the continuous decrease

Overall, derivative analysis confirms and complements the understanding of the graph and its decreasing intervals.