91Ó°ÊÓ

Vertical asymptotes are important features in the graph of a rational function. They occur where the function approaches infinity as the values of the input variable (x) get closer and closer to a specific number, causing the graph to spike upwards or downwards. This behavior happens at the values of x that make the denominator zero, because division by zero is undefined in mathematics. For the function \[ f(x) = \frac{x-1}{x^2-x-6} \]factor the denominator to set it equal to zero. Here, we decompose it as \[ (x-3)(x+2) = 0 \].So, the vertical asymptotes are located at \[ x = 3 \] and \[ x = -2 \].These lines will not touch the graph, but the curve will get infinitely close to them as it stretches towards infinity or negative infinity.

• Vertical asymptotes indicate a division by zero point.
• They are found by setting the denominator equal to zero and solving for x.
• Each zero of the denominator where the numerator is nonzero corresponds to a vertical asymptote.

Horizontal asymptotes describe the behavior of a rational function as the variable x moves towards positive or negative infinity. They show the value that the function approaches but never quite reaches, giving us an idea of the long-term trend of the graph.
For example, with \[ f(x) = \frac{x-1}{x^2-x-6} \],to determine the horizontal asymptote:

• Compare the degree of the numerator and the denominator.
• Here, the degree of the numerator (x-1) is 1, and the denominator (x^2-x-6) is 2.
• When the degree of the denominator is larger than the numerator, as in this case, the horizontal asymptote is located at y = 0.

The graph of this function will get closer to the x-axis as x becomes very large or very small, settling around y = 0.

The roots of a rational function are the x-values where the graph of the function crosses or touches the x-axis. To find these roots, we focus on the numerator of the function, setting it equal to zero and solving for x.
For the given rational function \[ f(x) = \frac{x-1}{x^2-x-6}, \] set the numerator \[ x-1 = 0 \] and solve to get \[ x = 1 \].This indicates that the graph crosses the x-axis at this point.

• Roots are found by setting the numerator equal to zero.
• The value of the function at the root is zero, indicating an x-intercept.
• Finding the roots helps us understand where the function has zero value or where it intersects the x-axis.

Knowing the roots not only provides x-intercepts but also plays a critical role when setting up the window for graphing, allowing better insight into important areas of interaction on the graph.