determine an equation for the graph to the right. the equation of the graph is y = . (type an expression…

1. Analyze the form of the rational - function graph:
   • The graph appears to be a rational function of the form (y=\frac{1}{(x - a)(x - b)}) (a hyperbola). The vertical asymptotes of a rational function (y = \frac{f(x)}{g(x)}) occur at the values of (x) for which (g(x)=0).
   • From the graph, the vertical asymptotes are (x=- 2) and (x = 2). So, the denominator of the rational function is ((x + 2)(x - 2)=x^{2}-4).
   • The general form of the rational function is (y=\frac{k}{x^{2}-4}), where (k) is a non - zero constant.
2. Find the value of the constant (k):
   • We can use a point on the graph to find the value of (k). The graph passes through the point ((0,-\frac{1}{4})).
   • Substitute (x = 0) and (y=-\frac{1}{4}) into the equation (y=\frac{k}{x^{2}-4}).
   • When (x = 0), we have (-\frac{1}{4}=\frac{k}{0^{2}-4}).
   • Simplify the right - hand side: (\frac{k}{-4}).
   • Then, by cross - multiplying, we get (-1\times(-4)=4k), so (k = 1).

Explanation:

Step1: Identify the denominator from vertical asymptotes

The vertical asymptotes are (x=-2) and (x = 2), so the denominator is ((x + 2)(x - 2)=x^{2}-4).

Step2: Determine the constant (k)

Substitute the point ((0,-\frac{1}{4})) into (y=\frac{k}{x^{2}-4}), solve for (k) and get (k = 1).

Answer:

(\frac{1}{x^{2}-4})