Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
4. Polynomial Functions
Understanding Polynomial Functions
9:05 minutes
Problem 45b
Textbook Question
Textbook QuestionIf the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a)domain (b)range (c)end behavior (d)number of zeros (e)number of turning points 10x7
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dominating Term in Polynomials
The dominating term of a polynomial function is the term with the highest degree, which significantly influences the function's behavior as the input values become very large or very small. For example, in the polynomial 10x^7, the dominating term is 10x^7, which dictates the overall shape and characteristics of the graph, especially in terms of end behavior.
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End Behavior of Polynomials
End behavior refers to the behavior of the graph of a polynomial function as the input values approach positive or negative infinity. The leading term determines whether the graph rises or falls at the ends; for instance, a positive leading coefficient with an odd degree, like 10x^7, indicates that the graph will rise to the right and fall to the left.
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Zeros and Turning Points
Zeros of a polynomial function are the x-values where the graph intersects the x-axis, while turning points are where the graph changes direction. The number of zeros can be at most equal to the degree of the polynomial, and the number of turning points is at most one less than the degree. For a polynomial of degree 7, like 10x^7, there can be up to 7 zeros and up to 6 turning points.
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