Follow the seven steps to graph each rational function. f(x)=x⁴/(x²+2)

Verified step by step guidance

1. Identify the domain of the function \(f(x) = \frac{x^{4}}{x^{2} + 2}\). Since the denominator is \(x^{2} + 2\), which is always positive for all real \(x\), the domain is all real numbers, \((-\infty, \infty)\).
2. Find the intercepts: For the y-intercept, evaluate \(f(0) = \frac{0^{4}}{0^{2} + 2}\). For the x-intercepts, set the numerator equal to zero, \(x^{4} = 0\), and solve for \(x\).
3. Determine the vertical asymptotes by finding values of \(x\) that make the denominator zero. Since \(x^{2} + 2 = 0\) has no real solutions, there are no vertical asymptotes.
4. Find the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator. The numerator degree is 4 and the denominator degree is 2. Since the numerator degree is greater, perform polynomial division of \(x^{4}\) by \(x^{2} + 2\) to find the oblique asymptote.
5. Analyze the end behavior of the function using the quotient from the polynomial division and sketch the graph accordingly, including the intercepts and asymptotes found.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the domain, zeros, and behavior of both numerator and denominator is essential for graphing and analyzing these functions.

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity, indicating important features in the graph.

End Behavior and Horizontal/Oblique Asymptotes

End behavior describes how the function behaves as x approaches infinity or negative infinity. For rational functions, this is determined by comparing the degrees of numerator and denominator, which helps identify horizontal or oblique asymptotes guiding the graph's long-term trend.