91Ó°ÊÓ

In three-dimensional surface graphs, understanding the asymptotic behavior of a function helps us predict how it behaves at extreme values. For the function \( f(x, y) = \frac{x^2}{y} \), this behavior is particularly significant due to its dependence on both \( x \) and \( y \).
As \( y \to 0^+ \), the surface approaches infinity, leading to a vertical asymptote. This means that as \( y \) becomes very small (positive), the value of \( f(x, y) \) becomes very large for any \( x eq 0 \). This represents a steep rise or spike in the surface.
Similarly, when \( |x| \to \infty \) with a fixed \( y > 0 \), the function value also approaches infinity. This describes the surface's tendency to become unbounded as \( x \) values increase in either direction. Thus, both ends in the \( x \)-direction suggest that the surface stretches upwards without bound, showing how the surface extends along these axes.

Cross-sections of a 3D surface graph help us see slices of the surface from different perspectives. For \( f(x, y) = \frac{x^2}{y} \), examining cross-sections gives clear insight into its shape.
When we take cross-sections for different constant values of \( y \), the graphs are parabolas. For instance, setting \( y = 1 \) involves looking at the equation \( f(x, 1) = x^2 \). This is a classic upward opening parabola, which indicates that these sections are not linear but curved.

• If \( y = c \), \( c > 0 \), the section is \( f(x, c) = \frac{x^2}{c} \). As \( c \) increases, the parabola gets wider. This is because the \( x^2 \) is divided by a larger value, flattening the parabola.

These cross-sections help you see how the surface’s height varies with different \( y \) values. Essentially, as \( y \) becomes larger, the parabola widens and flattens, clearly illustrating changes in the surface height.

Polynomial functions often appear in the form of sums of powers of variables. In our example, the function \( f(x, y) = \frac{x^2}{y} \) shows \( x^2 \), a simple polynomial expression in the variable \( x \).
This polynomial term \( x^2 \) fundamentally shapes the behavior of the surface in relation to \( x \). For fixed \( y \), the \( x^2 \) term defines parabolic cross-sections. When graphed, these parabolas capture the square relationship, explaining steep rises and the rapid increase in surface height as \( x \) moves away from the origin.

• Due to the power of 2, the graph is symmetric about the \( yz \)-plane because \( x^2 \) is always non-negative, causing equal behavior in both the positive and negative \( x \) directions.

Recognizing this polynomial structure in a function is crucial for understanding how its structure influences the resulting 3D graph's shape and symmetry.