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Show that f(x) = x^3 - 2x - 1 has a real zero between 1 and 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 17
In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>
Guida verificata passo dopo passo1
Identify the degree of the polynomial function. Since the function is \(f(x) = (x - 3)^2\), expand or recognize that the degree is 2 because the expression is squared.
Determine the leading coefficient. When expanded, the leading term is \(x^2\), so the leading coefficient is 1, which is positive.
Apply the Leading Coefficient Test: For an even degree polynomial with a positive leading coefficient, the end behavior is that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
Use this end behavior to match the polynomial with the correct graph. Look for a graph where both ends rise upwards, consistent with the behavior of a positive leading coefficient and even degree.
Confirm that the graph also reflects the vertex at \(x=3\), since the function is \((x-3)^2\), which shifts the parabola to the right by 3 units.
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Leading Coefficient Test
The Leading Coefficient Test helps determine the end behavior of a polynomial function by examining the degree and the leading coefficient. For large positive or negative values of x, the sign and degree dictate whether the graph rises or falls on each end.
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06:08
End Behavior of Polynomial Functions
Polynomial Degree and Its Effect on Graph Shape
The degree of a polynomial indicates the highest power of x and influences the number of turning points and the general shape of the graph. Even-degree polynomials have similar end behaviors on both sides, while odd-degree polynomials have opposite end behaviors.
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05:25Graphing Polynomial Functions
End Behavior of Polynomial Functions
End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. Understanding this helps match the function to its graph by predicting whether the graph rises or falls at the extremes.
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06:08End Behavior of Polynomial Functions
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