Problem 90 - Algebra and Trigonometry

Question

A Rational Function with No Asymptote Explain how you can tell (without graphing it) that the function $$ r(x)=\frac{x^{6}+10}{x^{4}+8 x^{2}+15} $$ has no \(x\) -intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?'

Step-by-Step Solution

Answer

No asymptotes exist; function behaves like \(x^2\) for large \(x\).

Step 1: Identify conditions for vertical asymptotes

A vertical asymptote occurs when the denominator of a rational function is equal to zero, and the numerator is not zero at those points. To test for vertical asymptotes in \( r(x) = \frac{x^6 + 10}{x^4 + 8x^2 + 15} \), consider the equation of the denominator: \( x^4 + 8x^2 + 15 = 0 \). This resembles a quadratic in terms of \( y = x^2 \): \( y^2 + 8y + 15 = 0 \). Solving for \( y \), use the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 1, b = 8, c = 15 \). The discriminant is \( 8^2 - 4 \cdot 1 \cdot 15 = 64 - 60 = 4 \), which is positive, giving real roots. Hence no values of \( x \) make the denominator zero without making the numerator zero as well, thus no vertical asymptote.

Step 2: Check for x-intercepts

To find \( x \)-intercepts, set the numerator \( x^6 + 10 = 0 \). Solving for \( x^6 = -10 \) results in a contradiction, as any real number raised to an even power is non-negative. Therefore, \( r(x) \) has no real \( x \)-intercepts.

Step 3: Assess for horizontal asymptotes

Horizontal asymptotes for a rational function occur by the degrees of the polynomials. If the degree of the numerator equals the degree of the denominator, the asymptote is \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients. In \( r(x) = \frac{x^6 + 10}{x^4 + 8x^2 + 15} \), the numerator has degree 6 and the denominator is degree 4, exceeding the denominator. This implies no horizontal asymptote.

Step 4: Consider slant asymptotes

A slant (oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In this function, since the numerator is of a higher degree than exactly one more, the condition doesn’t hold. Thus, \( r(x) \) has no slant asymptote.

Step 5: Determine end behavior

As \( x \to \infty \) or \( x \to -\infty \), the polynomial terms dominate rational functions. Here, the \( x^6 \) term in the numerator will significantly overshadow all others, and the same goes for \( x^4 \) in the denominator. Thus, the end behavior can be described by \( y = \frac{x^6}{x^4} = x^2 \), leading \( r(x) \to \infty \) as \( x \to \pm\infty \).

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never touches. In rational functions, vertical and horizontal asymptotes are common.

• Vertical asymptotes occur when the denominator of a rational function is zero, as long as the numerator isn't zero at that point too.
• Horizontal asymptotes depend on the degree of polynomials in both numerator and denominator.
• Slant or oblique asymptotes appear when the degree of the numerator is one more than that of the denominator.

For the function \( r(x) = \frac{x^6 + 10}{x^4 + 8x^2 + 15} \), examining the denominator, \( x^4 + 8x^2 + 15 \), for zero reveals no factors of the complete problem without also nullifying the numerator. As such, no vertical asymptote exists. Since the numerator's degree exceeds the denominator's by more than one, there are neither horizontal nor slant asymptotes in this case.

X-intercepts

X-intercepts occur where the graph crosses the x-axis, which happens when the numerator of a rational function equals zero. For \( r(x) = \frac{x^6 + 10}{x^4 + 8x^2 + 15} \), x-intercepts would be found by solving \( x^6 + 10 = 0 \). Any real number raised to an even power—like \( x^6 \)—cannot be negative, yet here it must equal \(-10\). This is not possible, meaning there are no real solutions and thus no x-intercepts exist for this function. The lack of real roots indicates that the graph remains above or below the x-axis for all x-values, depending on other factors.

End Behavior of Functions

The end behavior of rational functions describes how a graph behaves as \( x \) approaches infinity or negative infinity. Our function, \( r(x) = \frac{x^6 + 10}{x^4 + 8x^2 + 15} \), primarily shows its end behavior through its highest degree terms in the numerator and denominator.

• The highest power in the numerator, \( x^6 \), will dominate as \( x \to \infty \) or \( x \to -\infty \).
• In the denominator, the term \( x^4 \) is the highest power.

The behavior at infinity can be described by simplifying the fraction of these leading terms: \( \frac{x^6}{x^4} = x^2 \). As \( x \to \pm\infty \), \( r(x) \to \infty \), indicating the graph rises dramatically in both directions far from the origin, showing quadratic growth in terms of end behavior.

Polynomial Degrees in Rational Functions

The degree of a polynomial is the highest power of its variable. In rational functions, the relationship between the degrees of the numerator and denominator dictates several key attributes of the function.

• If the numerator and denominator have the same degree, there is a horizontal asymptote.
• If the numerator's degree is greater than the denominator's, as it is in our case, \( r(x) = \frac{x^6 + 10}{x^4 + 8x^2 + 15} \), then no horizontal asymptote exists, and the degree dictates a more complex end behavior.
• When the numerator exceeds the denominator by one, slant asymptotes might occur, but not here since \( x^{6} \) differs from \( x^{4} \) by more than simply one degree.

Recognizing that the numerator's degree is much higher (\(6 > 4\)), the overall control of the graph's behavior and lack of asymptotes comes directly from this unbalanced degree relationship.