Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{4}-5 x^{2}+4}{x^{4}-24 x^{2}+108}$$

Step-by-Step Solution

Answer

The graph has vertical asymptotes at \( x = \pm \sqrt{18} \) and \( x = \pm \sqrt{6} \), a horizontal asymptote at \( y = 1 \).

Step 1: Identify the Domain

To find the domain of \( f(x) \), set the denominator equal to zero and solve for \( x \): \[ x^4 - 24x^2 + 108 = 0 \]This is a quadratic in terms of \( x^2 \), so let \( u = x^2 \). The equation becomes: \[ u^2 - 24u + 108 = 0 \]Solve using the quadratic formula: \[ u = \frac{24 \pm \sqrt{24^2 - 4 \cdot 1 \cdot 108}}{2 \cdot 1} = \frac{24 \pm \sqrt{576 - 432}}{2} = \frac{24 \pm 12}{2} \]Which gives \( u = 18 \) or \( u = 6 \). Reverse substitution, \( x^2 = 18 \) or \( x^2 = 6 \), results in \( x = \pm \sqrt{18} \) or \( x = \pm \sqrt{6} \). These points are excluded from the domain.

Step 2: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. The denominator is zero at \( x = \pm \sqrt{18} \) and \( x = \pm \sqrt{6} \). Check the numerator:- \( x^2 = 18 \): \( x^4 - 5x^2 + 4 = (18)^2 - 5(18) + 4 eq 0 \)- \( x^2 = 6 \): \( x^4 - 5x^2 + 4 = (6)^2 - 5(6) + 4 eq 0 \)Thus, vertical asymptotes are at \( x = \pm \sqrt{18} \) and \( x = \pm \sqrt{6} \).

Step 3: Find Horizontal Asymptotes

Evaluate the degrees of the numerator and the denominator. Both are degree 4 polynomials, so divide the leading coefficients to find the horizontal asymptote: \[ y = \frac{1}{1} = 1 \]Thus, there is a horizontal asymptote at \( y = 1 \).

Step 4: Analyze End Behavior

As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 1 \) since the degrees of the numerator and denominator are equal, confirming the horizontal asymptote.

Step 5: Sketching the Graph

Include the asymptotes and domain restrictions on the graph:- Plot vertical asymptotes at \( x = \pm \sqrt{18} \) and \( x = \pm \sqrt{6} \).- Draw a horizontal asymptote along \( y = 1 \).- Ensure the curve approaches these lines appropriately but never crosses vertical asymptotes.

Key Concepts

Asymptotes

Asymptotes play a crucial role when analyzing rational functions. These are lines that the graph of the function approaches but never touches.
In the context of the exercise, we investigated vertical and horizontal asymptotes. Vertical asymptotes occur at the values of

• Vertical Asymptotes: These occur where the denominator of the function becomes zero, as long as the numerator does not become zero at those points. For the given function, vertical asymptotes are found at \(x = \pm \sqrt{18}\) and \(x = \pm \sqrt{6}\).
• Horizontal Asymptotes: These are found by comparing the degrees of the polynomial in the numerator and the denominator. Since both are degree 4, the horizontal asymptote is determined by dividing the leading coefficients. This tells us that the horizontal asymptote for this function is \(y = 1\).

Understanding these concepts helps in constructing a more accurate graph of the function by showing how the function behaves as it nears infinity or approaches undefined points.

Domain of Functions

The domain of a function is the set of all possible input values (\(x\)) for which the function is defined. For rational functions, the domain excludes any value that makes the denominator zero, as division by zero is undefined.
In our exercise, the function \(f(x) = \frac{x^{4}-5x^{2}+4}{x^{4}-24x^{2}+108}\) requires us to find where the denominator is zero.
By setting \(x^4 - 24x^2 + 108 = 0\), and solving, we found the values \(x = \pm \sqrt{18}\) and \(x = \pm \sqrt{6}\). These values are excluded from the domain. The function's domain is all real numbers except these values, ensuring the function remains defined.

End Behavior

End behavior describes how a function behaves as \(x\) approaches positive or negative infinity. For rational functions, this is closely linked with horizontal asymptotes.
For our given function, since the degrees of the numerator and the denominator are equal (both degree 4), the end behavior of the function will lead

• As \(x \to \infty\), \(f(x) \to 1\),
• Similarly, as \(x \to -\infty\), \(f(x) \to 1\).

This happens because the leading coefficients of the polynomial in the numerator and denominator are equal, resulting in the horizontal asymptote \(y = 1\). This provides insight into the long-term behavior of the graph and confirms the horizontal asymptote already identified.

Polynomial Functions

Rational functions are made up of polynomial functions in both the numerator and the denominator. Understanding polynomial behavior is fundamental to analyzing rational functions.
In this exercise, both the numerator \(x^4 - 5x^2 + 4\) and the denominator \(x^4 - 24x^2 + 108\) are fourth-degree polynomials. This high degree suggests several features:

• Roots: Polynomials may have several roots (zeros) which affect the overall shape and intercepts of the graph of the rational function.
• Leading Coefficient: The leading coefficient in both the numerator and denominator is 1. When degrees are equal, it determines the horizontal asymptote.

Grasping these characteristics helps ensure a clearer understanding of how the rational function behaves, enhancing the efficiency in graph sketching and function analysis.