91Ó°ÊÓ

Rational functions are an important concept in mathematics and require a solid understanding to work with effectively. A rational function is a fraction where both the numerator and the denominator are polynomials. The function can be expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). When analyzing rational functions, it’s critical to consider the behavior of both the numerator and the denominator.

• The numerator, \( P(x) \), influences the zeros of the function, which are points where the function crosses the x-axis.
• The denominator, \( Q(x) \), influences the vertical asymptotes, but only when it equals zero and the numerator does not also equal zero at that point.

For the given exercise, the rational function is \( y = \frac{-15}{x^2+x+4} \). Notice how the numerator is a constant, which means the function does not cross the x-axis unless the numerator itself becomes zero, which isn't the case here. The denominator has no real roots (does not factor neatly over the reals as determined by the quadratic equations discriminant, which is negative), so there are no vertical asymptotes.

Graphing is a powerful tool to visualize the behavior of rational functions. It allows us to see trends, patterns, and unusual behavior that might not be evident through algebraic manipulation alone.
When graphing \( y = \frac{-15}{x^2 + x + 4} \), consider these points:

• The curve of this function will be a type of hyperbola.
• Start by plotting the intercepts, if they exist; dynamics around the axes can reveal key traits about the graph.
• A graph won't pass through the x-axis because the numerator isn't zero. However, check behavior near the y-axis by substituting \( x = 0 \).
• Determine curvature by considering the end behavior with horizontal asymptotes.

In our function, since there are no roots in the denominator, the curve does not touch or intersect the x-axis. Evaluating at \( x = 0 \) gives \( y = -\frac{15}{4} \), which is an anchor point for your sketch.

Asymptotes provide critical insights into the limiting behavior of functions as they approach certain points. With rational functions, we typically consider vertical, horizontal, and oblique asymptotes. Here, we focus on horizontal asymptotes:

• Vertical asymptotes are missing here due to the lack of real roots of the denominator.
• A horizontal asymptote occurs at \( y = 0 \) for the final function \( y = \frac{-15}{x^2 + x + 4} \) because as \( x \) approaches infinity or negative infinity, the degree of the polynomial in the denominator is greater than that in the numerator.

The presence of a horizontal asymptote at \( y = 0 \) means that as \( x \) moves away from 0 in either direction, the value of \( y \) gets closer to zero, but never actually reaches it. This tells us about the end behavior of the graph and helps guide where to shade when determining solutions to inequalities like \( y > \frac{-15}{x^2 + x + 4} \). The shaded region will be above this asymptote for large or small x-values.