Problem 18 - Precalculus with Limits

Question

In Exercises 5-18, sketch the graph of the inequality. $$y>\frac{-15}{x^{2}+x+4}$$

Step-by-Step Solution

Answer

The steps to solve this problem include drawing the graph of the function, determining the region that satisfies the inequality, and shading this region. The solution is a graphical representation that depicts all y-values above the curve of the function \(y = -15 / (x^2 + x + 4)\), which satisfy the inequality.

Step 1: Draw the Graph of the Function

The first step is to plot the graph of the equation \(y = -15 / (x^2 + x + 4)\). This is a rational function and its graph will be a hyperbola. There are no vertical asymptotes because the polynomial in the denominator has no real roots. There is a horizontal asymptote at \(y=0\).\nIn the process of creating this graph, take note of all the critical points where the graph intersects the x-axis, y-axis, or any noticeable points of inflection or curvature.

Step 2: Determine the Solution Region

Now, one has to identify the region which satisfies the inequality \(y > -15 / (x^2 + x + 4)\). This is the region where the values of y would be greater than the function.\nSo, clearly all the y-values above the curve of the function will satisfy the inequality.

Step 3: Shade the Solution Region

The final step is to shade the region representing the solution of the inequality, which would be all the y-values above the curve of the function. This would visually represent all values of x and y that satisfy the inequality.

Key Concepts

Understanding Rational Functions

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.

Visualizing with Graphing

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.

Identifying Asymptotes

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.