Textbook Question
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 4, Problem 11
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).]
Verified step by step guidance1
Identify the leading term of the polynomial function. For the given function \(f(x) = x^6 - 6x^4 + 9x^2\), the leading term is \(x^6\) because it has the highest power of \(x\).
Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 6, an even number.
Look at the leading coefficient, which is the coefficient of the leading term. In this case, it is 1 (positive).
Apply the Leading Coefficient Test: For an even degree polynomial with a positive leading coefficient, the end behavior is such that as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
Use this end behavior information to match the polynomial function with the graph that shows both ends of the graph rising upwards.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 even-degree polynomials with a positive leading coefficient, both ends of the graph rise; if negative, both fall. For odd-degree polynomials, the ends go in opposite directions depending on the sign of the leading coefficient.
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End Behavior of Polynomial Functions
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the expression. It influences the shape and end behavior of the graph. In this problem, the polynomial degree is 6, an even number, which affects how the graph behaves as x approaches positive or negative infinity.
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05:16Standard Form of Polynomials
End Behavior of Polynomial Functions
End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It is determined by the leading term and helps match the polynomial to its graph by predicting whether the graph rises or falls on each end.
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06:08End Behavior of Polynomial Functions
Related Practice
Textbook Question
In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x3−3x2−11x+6
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Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As ______
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z.
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Textbook Question
Identify which graphs are not those of polynomial functions.
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Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (4x4−4x2+6x)/(x−4)
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