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
Identify which graphs are not those of polynomial functions.
Verified step by step guidance1
Step 1: Understand the characteristics of polynomial function graphs. Polynomial graphs are smooth and continuous curves without breaks, holes, or sharp corners.
Step 2: Observe the given graph carefully. Notice that the curve is smooth and continuous, with no breaks or sharp turns.
Step 3: Identify the general shape of the graph. The graph appears to be a parabola opening downward, which is typical of a quadratic polynomial function of the form \(y = ax^2 + bx + c\) where \(a < 0\).
Step 4: Confirm that the graph does not have any features that contradict polynomial behavior, such as vertical asymptotes or discontinuities.
Step 5: Conclude that since the graph is smooth, continuous, and resembles a parabola, it is consistent with the graph of a polynomial function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions
Polynomial functions are algebraic expressions involving variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Their graphs are smooth, continuous curves without breaks or sharp corners, and they extend infinitely in both directions.
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Graph Characteristics of Polynomial Functions
Graphs of polynomial functions are continuous and smooth, with no gaps or cusps. They can have turning points, but the number of turning points is limited by the degree of the polynomial. The end behavior of the graph depends on the leading term's degree and coefficient.
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Identifying Non-Polynomial Graphs
Graphs that are not polynomial functions often have discontinuities, sharp corners, vertical asymptotes, or oscillations that do not smooth out. Recognizing these features helps distinguish polynomial graphs from others like rational, exponential, or piecewise functions.
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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 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).]
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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.
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)=x3−2x2−11x+12
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Textbook Question
Divide using long division. State the quotient, and the remainder, r(x). (4x4−4x2+6x)/(x−4)