Textbook Question
Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = -3x^2 + 18x + 1
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4. Polynomial Functions
Understanding Polynomial Functions
2:54 minutesProblem 46
Textbook Question
If 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-9x6
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Identify the dominating term of the polynomial function, which is \(-9x^6\).
For the domain of the polynomial function, note that polynomial functions are defined for all real numbers, so the domain is \((-\infty, \infty)\).
For the range, consider that the leading term \(-9x^6\) is an even degree term with a negative coefficient, which means the graph opens downwards, and the range is \((-\infty, 0]\).
For the end behavior, as \(x\) approaches \(\pm\infty\), the term \(-9x^6\) dominates, so the function \(f(x)\) approaches \(-\infty\).
For the number of zeros and turning points, note that a polynomial of degree 6 can have up to 6 real zeros and up to 5 turning points.
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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 is the term with the highest degree, which significantly influences the polynomial's behavior as the input values become very large or very small. In the case of the polynomial -9x^6, the dominating term is -9x^6, indicating that the function will exhibit specific characteristics based on the degree and leading coefficient.
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End Behavior of Polynomials
End behavior describes how the values of a polynomial function behave as the input approaches positive or negative infinity. For the polynomial -9x^6, since the leading term has an even degree and a negative coefficient, the ends of the graph will both point downwards, indicating that the function approaches negative infinity in both directions.
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Number of Zeros and Turning Points
The number of zeros of a polynomial function corresponds to the x-intercepts of its graph, while turning points are where the graph changes direction. For a polynomial of degree 6, like -9x^6, it can have up to 6 zeros and up to 5 turning points, but the actual number may vary based on the specific polynomial's factors and behavior.
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