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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Chapter 4, Problem 27
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=3+2x-4x2-5x10
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Identify the degree and leading coefficient of the polynomial function. The given function is \(f(x) = 3 + 2x - 4x^2 - 5x^{10}\). The term with the highest power of \(x\) is \(-5x^{10}\), so the degree is 10 and the leading coefficient is \(-5\).
Determine the end behavior based on the degree and leading coefficient. Since the degree is even (10) and the leading coefficient is negative (\(-5\)), the ends of the graph will both point downwards.
Express the end behavior in terms of limits: As \(x \to \infty\), \(f(x) \to -\infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).
Draw or visualize the end behavior diagram: both ends of the graph go down towards negative infinity.
Summarize the end behavior: The graph falls to negative infinity on both the left and right ends because the leading term dominates the behavior for very large positive and negative values of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions and Degree
A polynomial function is an expression consisting of variables raised to whole-number exponents and their coefficients. The degree of the polynomial is the highest exponent of the variable, which largely determines the shape and end behavior of its graph.
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Introduction to Polynomial Functions
Leading Term and Leading Coefficient
The leading term of a polynomial is the term with the highest degree, and its coefficient is the leading coefficient. These determine the end behavior of the polynomial's graph, indicating how the function behaves as x approaches positive or negative infinity.
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06:08End Behavior of Polynomial Functions
End Behavior of Polynomial Graphs
End behavior describes how the values of a polynomial function behave as x approaches infinity or negative infinity. It depends on the degree and leading coefficient: even-degree polynomials with positive leading coefficients rise on both ends, while odd-degree polynomials have opposite end behaviors.
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06:08End Behavior of Polynomial Functions
Related Practice
Textbook Question
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=7+2x-5x2-10x4
Textbook Question
Solve each polynomial inequality. Give the solution set in interval notation. (x - 4)(2x + 3)(3x - 1) ≥ 0
Textbook Question
Circumference of a Circle The circumference of a circle varies directly as the radius. A circle with radius 7 in. has circumference 43.96 in. Find the circumference of the circle if the radius changes to 11 in.
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
Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = -3 (x - 2)2 +1