Analyzing the Graph of a Polynomial

In pre-calculus, we investigate many topics in the subject, one of which is graphing polynomials.

Above is the graph of the equation (x-2)/(x2-2x-8). Using the graph, we can analyze many characteristics of the polynomial. First, we can identify the y and the x-intercepts, when x and y equal zero, respectively.

We can also recognize the vertical asymptotes, which is the value of x that the graph never touches. In this case, the vertical asymptotes are when x equals to -2 and 4. We can also describe the behavior of the graph as x approaches the vertical asymptote from different directions. For instance, as x approaches -2 from the negative side (left), y approaches negative infinity (-∞), and as x approaches -2 from the positive side (right), y approaches positive infinity (+∞).  The same behavior applies to when x approaches 4 from both directions.

The horizontal asymptote can also be found using this graph. As seen above, the graph never touches the point y equals 0. The behavior of the graph can be observed as y approaches zero from the negative (below) and positive (above) side. In this case, as y approaches zero from below, x approaches -∞ and as y approaches zero from above, x approaches +∞.

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