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When graphing rational functions like \(g(x) = \frac{2}{x^2}\), a table of values is an essential tool. By selecting diverse \(x\) values and computing the corresponding \(g(x)\), you can see how the function behaves across different sections of the graph. It's important to choose a mix of negative, positive, and zero for \(x\). Each will give insight into the symmetry and progression of the curve.

Here's how you do it:

• Choose \(x\) values like -3, -2, -1, 0, 1, 2, and 3.
• Calculate \(g(x)\) for each \(x\). Notice that \(g(0)\) is undefined because you cannot divide by zero.
• Record each point: (-3, \(\frac{2}{9}\)), (-2, \(\frac{1}{2}\)), (-1, 2), (0, undefined), (1, 2), (2, \(\frac{1}{2}\)), and (3, \(\frac{2}{9}\)).

This table helps lay the groundwork for plotting the graph.

Vertical asymptotes in rational functions occur where the function is undefined. For \(g(x) = \frac{2}{x^2}\), this is clearly shown at \(x = 0\). As \(x\) gets closer to zero, the function values tend to increase or decrease without bound, creating a vertical line that the graph approaches but never touches.

Understanding vertical asymptotes helps visualize how the function behaves near these points. Since division by zero is undefined, at \(x = 0\), \(g(x)\) doesn't have a value, creating a gap in the graph at this line.

Symmetry provides valuable insights into graph characteristics. The function \(g(x) = \frac{2}{x^2}\) is symmetric about the y-axis. This means that for every \(x\), \(g(-x)\) is equal to \(g(x)\). Consider:

• \(g(-1) = 2\) and \(g(1) = 2\)
• \(g(-2) = \frac{1}{2}\) and \(g(2) = \frac{1}{2}\)
• \(g(-3) = \frac{2}{9}\) and \(g(3) = \frac{2}{9}\)

This symmetry simplifies the graphing process since it's only necessary to calculate points for positive \(x\) values, and then mirror them over the y-axis.

Understanding a function's behavior at infinity helps predict its end behavior. For \(g(x) = \frac{2}{x^2}\), as \(x\) becomes very large or very small (approaching \(\pm \infty\)), the value of \(g(x)\) approaches zero. This situation creates a horizontal asymptote along the x-axis.

This behavior is typical in rational expressions where the degree of the polynomial in the denominator is greater than in the numerator. As \(x\) extends towards infinity, the effect of squaring \(x\) in the denominator dominates the expression, pulling \(g(x)\) closer and closer to zero. This knowledge helps predict the horizontal trend of the graph, ensuring an accurate sketch of the function's overall shape and direction.