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Understanding asymptotes is crucial when analyzing function graphs. Asymptotes are lines that a graph approaches but never actually touches. They can be either vertical, horizontal, or oblique (slant). In the function \( f(x) = \frac{-64}{x^2 + 16} \), we need to identify whether there are asymptotes and their type.
- **Vertical Asymptotes**: These occur when the denominator equals zero, causing the function to become undefined. However, for our equation, \( x^2 + 16 \) is never zero because any real number squared is non-negative and adding 16 keeps it positive. Hence, there are no vertical asymptotes.
- **Horizontal Asymptotes**: These are determined by the behavior of the function as \( x \) approaches positive or negative infinity. Since the degree of the polynomial in the denominator (2) is larger than the numerator (0), \( f(x) \) approaches zero as \( x \to \pm \infty \). Therefore, there is a horizontal asymptote at \( y = 0 \).
Identifying these lines can help determine the overall shape and behavior of the graph.

The domain of a function represents all the possible input values \( x \) for which the function is defined. For \( f(x) = \frac{-64}{x^2 + 16} \), we have seen that the denominator is always positive. There are no restrictions on the values that \( x \) can take, meaning the domain is all real numbers: \( x \in \mathbb{R} \).
The range is the set of possible output values \( y \). To find out the range, we consider the behavior of the function over its domain. As the denominator \( x^2 + 16 \) increases for any \( x \), the whole fraction decreases in magnitude since \(-64\) is a constant numerator. The largest value the function attains is \(-4\) when \( x = 0 \), because \( f(0) = -4 \). For any other \( x \), \( f(x) \) is less than \(-4\), approaching zero.
Thus, the range of the function can be defined as \( (-\infty, -4] \). This reflects how tightly the graph is bound below by the curve's behavior.

Symmetry in graphs helps simplify the understanding of graph behaviors and characteristics. A function is termed symmetric concerning the y-axis if \( f(x) = f(-x) \) for all x in its domain. This type of symmetry indicates that the function is even.
For the given function \( f(x) = \frac{-64}{x^2 + 16} \), we can test for symmetry by substituting \(-x\) in place of \(x\). The function becomes \( \frac{-64}{(-x)^2 + 16} \), which simplifies back to the original function \( \frac{-64}{x^2 + 16} \). Since \( f(x) = f(-x) \), the graph is symmetric about the y-axis.
Understanding that the curve has y-axis symmetry can aid in predicting its shape. For instance, when plotting points for the graph, it’s sufficient to compute values for positive \( x \), since the negative \( x \) values will mirror them. This insight into symmetry makes graph sketching an easier endeavor.