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Chapter 3: Problem 32

Determine the end behavior of the graph of the function. \(k(x)=11 x^{7}-4 x^{2}+9 x+3\)

Short Answer

Expert verified
As \(x\) goes to \(+\text{\textiunfty}\), \(k(x)\) goes to \(+\text{\textiunfty}\). As \(x\) goes to \(-\text{\textiunfty}\), \(k(x)\) goes to \(-\text{\textiunfty}\).

Step by step solution

01

Identify the Leading Term

The leading term in the polynomial function is the term with the highest power of x. For the function \(k(x) = 11x^7 - 4x^2 + 9x + 3\), the leading term is \(11x^7\).
02

Determine the Degree and Leading Coefficient

The degree of the polynomial is the highest power of x, which is 7. The leading coefficient is the coefficient of the leading term, which is 11.
03

Analyze the Leading Term's End Behavior

The degree is odd (7), and the leading coefficient is positive (11). For large values of \(x\), this means the end behavior of \(k(x)\) will be similar to \(y = 11x^7\).
04

Determine Behavior as x Approaches Infinity

As \(x\) approaches positive infinity, \(k(x)\) will also approach positive infinity because \(11x^7\) grows without bound in the positive direction.
05

Determine Behavior as x Approaches Negative Infinity

As \(x\) approaches negative infinity, \(k(x)\) will approach negative infinity because \(11x^7\) grows without bound in the negative direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Leading Term
The leading term in a polynomial function is crucial. It's the term with the highest power of the variable, usually denoted as the term with the largest exponent. In the polynomial function provided, The leading term tells us about the general shape and end behavior of the graph. Let's re-examine the given polynomial: The leading term, especially when identifying the end behavior of a polynomial function, oversimplifies the complex equation for easier understanding. Knowing the leading term simplifies analysis significantly.
Degree of Polynomial
The degree of a polynomial is what determines the polynomial's highest exponent among the terms. The degree reflects the term that holds the majority sway over the function's end behavior. For instance, the degree of the polynomial The degree acts as a quick reference for predicting graph behavior. Additionally, it informs us about the maximum number of roots or solutions a polynomial might possess. In the current function example, and understanding its implications eases the interpretation significantly.
Leading Coefficient
The leading coefficient is another key player in understanding polynomials. It's defined as the coefficient of the term with the highest power. This leading coefficient impacts the width and direction of a polynomial's graph. In our example, - a positive leading coefficient tips us off that the function will rise towards positive infinity on both ends, which is crucial for predicting graph trajectories. Also, if the leading coefficient were negative, we'd expect inverse behavior.
Polynomial End Behavior
Finally, let's dive into polynomial end behavior. The end behavior of a polynomial function describes what happens to the function's graph as the variable approaches positive or negative infinity. This concept heavily relies on understanding the leading term and its dynamics. For our function - primarily influenced by the odd degree and the positive leading coefficient. The best way to summarize the end behavior here is:
  • As x approaches positive infinity, our function also reaches positive infinity.
  • As x approaches negative infinity, our function hits negative infinity due to its odd degree.

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