Complete the following steps for the given functions.

c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f (x) = \frac{4 x^{3} + 4 x^{2} + 7 x + 4}{x^{2} + 1}$

검증된 단계별 안내

Identify the function: $ f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} $. This is a rational function where the degree of the numerator is higher than the degree of the denominator.

Determine the vertical asymptotes by setting the denominator equal to zero: $ x^2 + 1 = 0 $. Since this equation has no real solutions, there are no vertical asymptotes.

Find the horizontal or oblique asymptote. Since the degree of the numerator (3) is greater than the degree of the denominator (2), there is no horizontal asymptote. Instead, perform polynomial long division to find the oblique asymptote.

Perform polynomial long division of $ 4x^3 + 4x^2 + 7x + 4 $ by $ x^2 + 1 $ to find the quotient, which represents the oblique asymptote.

Use a graphing utility to plot the function $ f(x) $ and the oblique asymptote. Then, sketch the graph by hand, ensuring to correct any discrepancies observed in the computer-generated graph.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They can be vertical, horizontal, or oblique. Vertical asymptotes occur where the function is undefined, typically at values that make the denominator zero. Horizontal asymptotes indicate the behavior of the function as x approaches infinity, showing the end behavior of the graph.

Graphing Rational Functions

Graphing rational functions involves plotting the function defined as the ratio of two polynomials. Key steps include identifying intercepts, asymptotes, and the behavior of the function at critical points. Understanding the degree of the numerator and denominator helps predict the end behavior and the presence of horizontal asymptotes.

Error Correction in Graphing

Error correction in graphing involves comparing a computer-generated graph with a hand-drawn sketch to identify discrepancies. This process requires understanding the function's characteristics, such as asymptotes and intercepts, to ensure accuracy. By analyzing the graph's behavior at critical points, one can refine the sketch to better represent the function.