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Chapter 3: Problem 727

Determine the end behavior of the polynomial function. $$f(x)=8 x^{3}-3 x^{2}+2 x-4$$

Short Answer

Expert verified
The end behavior is: as \(x \to +\infty\), \(f(x) \to +\infty\); as \(x \to -\infty\), \(f(x) \to -\infty\).

Step by step solution

01

Identify the Leading Term

The end behavior of a polynomial function is determined by its leading term. The leading term of the polynomial function \(f(x) = 8x^3 - 3x^2 + 2x - 4\) is \(8x^3\) since it has the highest power of \(x\).
02

Determine the Degree and Leading Coefficient

The degree of the leading term \(8x^3\) is 3, and the leading coefficient is 8. Since the degree is odd and the leading coefficient is positive, it will determine the end behavior of the function.
03

Analyze End Behavior Based on Leading Term

For polynomials with an odd degree and a positive leading coefficient, the end behavior is that as \(x\) approaches \(+\infty\), \(f(x)\) approaches \(+\infty\). Similarly, as \(x\) approaches \(-\infty\), \(f(x)\) approaches \(-\infty\).

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Key Concepts

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

End Behavior
When we talk about the end behavior of a polynomial function, we are describing the direction in which the graph of the function extends as the input variable \(x\) moves towards positive infinity \((+\infty)\) or negative infinity \((-\infty)\). This behavior helps us understand the overall shape of the polynomial graphed on a coordinate plane. The leading term, which is the term with the highest power of \(x\), determines the end behavior of the polynomial. This is because, as \(|x|\) becomes very large, the highest power of \(x\) will dominate the other terms, making them almost negligible.
Leading Term
The leading term of a polynomial function is the term which has the highest exponent for the variable \(x\). It is essential because it determines how the polynomial behaves for very large or small values of \(x\). For instance, in the polynomial \(f(x) = 8x^3 - 3x^2 + 2x - 4\), the leading term is \(8x^3\). This term is what you focus on when determining the end behavior of the polynomial because as \(x\) grows very large (positively or negatively), the influence of all other terms fades in comparison to the leading term. This dominance forms the core reason why the leading term is crucial in predicting the end behavior.
Degree of a Polynomial
The degree of a polynomial is defined as the largest exponent among its terms. It tells us a lot about the nature and behavior of the function. For the polynomial \(f(x) = 8x^3 - 3x^2 + 2x - 4\), the degree is 3 because the highest power of \(x\) in the polynomial is \(x^3\).
  • If the degree is even, the ends of the graph will go in the same direction.
  • If the degree is odd, the ends of the graph will go in opposite directions.
This characteristic of the degree of a polynomial is pivotal in understanding not just its end behavior but also the number of turns the polynomial's graph can have depending on its value.
Leading Coefficient
The leading coefficient is the numerical coefficient of the term with the highest degree in a polynomial. It plays a significant role in determining the direction of the end behavior of the polynomial. In our function \(f(x) = 8x^3 - 3x^2 + 2x - 4\), the leading coefficient is 8.
  • If the leading coefficient is positive, for odd-degree polynomials, the graph will extend upwards on the right end \((x \to +\infty)\) and downwards on the left end \((x \to -\infty)\).
  • If it were negative, the directions would be reversed, meaning the graph would go downwards on the right end and upwards on the left end.
This detail is crucial because combining the sign of the leading coefficient with the degree aids in accurately predicting the direction of a polynomial graph's 'arms' as they shoot off into infinity.

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Most popular questions from this chapter

Use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $$x-2,4 x^{4}-15 x^{2}-4$$

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies inversely with \(x\) and when \(x=6, \quad y=2.\)

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the cube of \(z\) and inversely as the square root of \(w .\) When \(x=2, z=2,\) and \(w=64,\) then \(y=12 .\) Find \(y\) when \(x=1, z=3,\) and \(w=4\) .

For the following exercises, use long division to find the quotient and remainder. $$\frac{3 x^{4}-4 x^{2}+4 x+8}{x+1}$$

Use a calculator with CAS to answer the questions. Consider \(\frac{x^{4}-k^{4}}{x-k}\) for \(k=1, 2, 3,\) What do you expect the result to be if \(k=4 ?\)
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