Question

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(x^{2} y=4\)

Short answer

The graph of \(y = \frac{4}{x^{2}}\) has x = 0 and y = 0 as vertical and horizontal asymptotes respectively. The graph doesn't intersect with any of the axes, has no extrema and is symmetric about the y-axis.

Step-by-step solution

Express the equation in the form \(y = f(x)\)

Rewrite the equation \(x^{2} y=4\) in the form \(y=f(x)\) for easier analysis and graphing. Thus, we have \(y = \frac{4}{x^{2}}\).

Identify the intercepts

The x-intercepts are the values of x for which y is 0. Since the denominator of the function \(y = \frac{4}{x^{2}}\) cannot be 0, there are no x-intercepts. The y-intercept is the value of y when x is 0. However, this function is undefined at x=0, so there is no y-intercept.

Identify the asymptotes

A vertical asymptote occurs where the function is undefined or approaches an infinite value. As x approaches 0 from either the positive or negative direction, the function \(y = \frac{4}{x^{2}}\) will approach infinity. Hence x = 0 is a vertical asymptote. A horizontal asymptote is a line to which a graph approaches as x goes to infinity or negative infinity. In this case, as x approaches infinity or negative infinity, y will approach 0. Hence y = 0 is a horizontal asymptote.

Sketch the graph

Plot the asymptotes as dashed lines on the graph. The function \(y = \frac{4}{x^{2}}\) will approach but never cross these lines. The graph is symmetrical about the y-axis because squaring x eliminates any negative sign. Plotting a few points for x can give an idea of where the plot should pass. Note that the function has no extrema as there are no peaks or valleys.

Key concepts

Intercepts in Graphs of Rational Functions

Understanding intercepts is crucial for graphing rational functions. Intercepts are the points where a graph crosses the x-axis and y-axis. For a graph defined by a function written in the form of a fraction or ratio, these intercepts can provide valuable information about the function's behavior:

• X-intercepts: These occur where the function's value is zero. In simpler terms, it's where the graph touches or crosses the x-axis. For our function, however, since the denominator cannot be zero, there are no x-intercepts.
• Y-intercepts: These occur at the point where a line or curve crosses the y-axis, meaning x must be zero. Often, rational functions are undefined at x equals zero, as is the case with our function, which leads to no y-intercepts.

Identifying intercepts helps in giving a more complete picture of how the graph behaves as it moves across different axes.

Asymptotes: Guidelines for Rational Functions

Asymptotes are invisible lines that a function approaches but never actually meets. They are essential when sketching the graph of rational functions, as they dictate the behavior and direction of the graph at extreme points.

• Vertical Asymptotes: These occur where the function is undefined, often relating to the denominator being zero. For the function \(y = \frac{4}{x^{2}}\), the graph approaches infinity as x approaches zero from both sides, suggesting a vertical asymptote at \(x = 0\).
• Horizontal Asymptotes: These occur as x values become very large (either positive or negative). They indicate a line that the graph settles near. In our function, as x approaches both positive and negative infinity, y approaches zero, indicating a horizontal asymptote of \(y = 0\).

Plotting these asymptotes using dashed lines can help visualize and properly draw the graph without crossing these invisible boundaries.

Understanding Symmetry in Rational Functions

Symmetry can greatly simplify the process of graphing functions by reducing the amount of calculation and plotting needed. For rational functions, symmetry often occurs about certain axes due to the nature of their mathematical expressions.

The function \(y = \frac{4}{x^{2}}\) demonstrates symmetry about the y-axis. This means that the left side of the graph mirrors the right side. This kind of symmetry can occur when substituting \(-x\) for \(x\) in the function yields the original equation. In other words, it shows that for every point \((x, y)\), there is an equivalent point \((-x, y)\) on the graph.

Recognizing symmetry helps in sketching graphs quickly as you need only calculate one side and then mirror it. It provides a clearer picture and reduces potential errors during plotting.