Textbook Question
Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 5)2 - 4
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 26
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=10x6-x5+2x-2
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Identify the leading term of the polynomial function. For the given function \(f(x) = 10x^6 - x^5 + 2x - 2\), the leading term is \$10x^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 10, a positive number.
Use the degree and leading coefficient to describe the end behavior: For an even degree and positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
Summarize the end behavior using an end behavior diagram or notation: both ends of the graph rise upwards, which can be represented as \(\uparrow \quad \quad \uparrow\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 mainly by the leading term, which dominates the function for very large or very small x-values.
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End Behavior of Polynomial Functions
Leading Term and Degree of a Polynomial
The leading term is the term with the highest power of x in a polynomial. The degree (the exponent of this term) and the leading coefficient (its coefficient) dictate the shape and direction of the graph's ends.
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Guided course
05:16Standard Form of Polynomials
End Behavior Diagrams
End behavior diagrams use arrows or symbols to visually represent the direction of the graph as x approaches ±∞. They help summarize whether the graph rises or falls on each end based on the polynomial's degree and leading coefficient.
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06:08End Behavior of Polynomial Functions
Related Practice
Textbook Question
Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. -2+√5, -2-√5, -2, 1
Textbook Question
Solve each polynomial inequality. Give the solution set in interval notation. (x - 4)(2x + 3)(3x - 1) ≥ 0
Textbook Question
Factor ƒ(x) into linear factors given that k is a zero.
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Textbook Question
Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = -(1/2)(x + 1)2 - 3
Textbook Question
Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x) = (x-k) q(x) + r. ƒ(x) = 2x3 + 3x2 - 16x+10; k = -4