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Follow the seven steps to graph each rational function. f(x)=2/(x2+x−2)
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 69
Follow the seven steps to graph each rational function. f(x)=2x2/(x2+4)
Guida verificata passo dopo passo1
Identify the domain of the function by finding all values of \(x\) for which the denominator is not zero. For \(f(x) = \frac{2x^{2}}{x^{2} + 4}\), set the denominator equal to zero: \(x^{2} + 4 = 0\) and solve for \(x\).
Find the intercepts: To find the \(y\)-intercept, evaluate \(f(0)\). To find the \(x\)-intercepts, set the numerator equal to zero and solve for \(x\).
Determine any vertical asymptotes by analyzing where the denominator is zero and the numerator is not zero. Since the denominator is \(x^{2} + 4\), check if it has real roots.
Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials. Since both numerator and denominator are degree 2, divide the leading coefficients to find the horizontal asymptote.
Analyze the behavior of the function near the asymptotes and at large values of \(x\) to understand the end behavior, then sketch the graph using all the information gathered.
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Rational Functions
A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined values and to analyze the function's behavior.
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6:04
Intro to Rational Functions
Graphing Steps for Rational Functions
Graphing rational functions involves seven key steps: finding the domain, intercepts, asymptotes (vertical, horizontal, or oblique), analyzing end behavior, plotting points, and sketching the curve. These steps help visualize the function accurately.
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8:19How to Graph Rational Functions
Asymptotes and End Behavior
Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe the function's behavior as x approaches infinity or negative infinity.
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6:24Introduction to Asymptotes
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