Question

What are the end behaviors of $$ y=3 x^3+15 x^2-6 x+5 ? $$ A. \(+\infty\) for \(x<0\) and \(-\infty\) for \(x>0\) B. \(-\infty\) for \(x<0\) and \(-\infty\) for \(x>0\) C. \(-\infty\) for \(x<0\) and \(+\infty\) for \(x>0\) D. \(+\infty\) for \(x<0\) and \(+\infty\) for \(x>0\)

Short answer

C. \(-\infty\) for \(x < 0\) and \(+\infty\) for \(x > 0\)

Step-by-step solution

Identify the Leading Term

The leading term is the term with the highest power of x. For the function \(y = 3x^3 + 15x^2 - 6x + 5\), the leading term is \(3x^3\).

Determine the Degree and Leading Coefficient

The degree of the polynomial is the highest power of x in the leading term, which is 3. The leading coefficient is the coefficient of the leading term, which is 3.

Analyze the End Behavior Based on the Degree and Leading Coefficient

For a polynomial of degree 3 (an odd degree) with a positive leading coefficient, the end behaviors are as follows: - As \(x \rightarrow -\infty\), \(y \rightarrow -\infty\) - As \(x \rightarrow +\infty\), \(y \rightarrow +\infty\)

Match the End Behaviors to the Given Choices

Comparing the analyzed end behaviors to the provided choices, only one option matches: - \(-\infty\) for \(x < 0\) and \(+\infty\) for \(x > 0\), which corresponds to option C.

Key concepts

Leading Term

In a polynomial, the leading term is the term with the highest power of the variable, which in this case is \(x\). Identifying the leading term is the first step in analyzing the behavior of the polynomial. For the polynomial given:

• y = 3x^3 + 15x^2 - 6x + 5

The leading term is \(3x^3\) as it contains \(x\) raised to the highest power, which is \(3\). This term is important because it largely determines the end behavior of the polynomial.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable \(x\). In the given polynomial:

• y = 3x^3 + 15x^2 - 6x + 5

The degree is \(3\) because the leading term \(3x^3\) contains \(x\) to the power of \(3\). The degree helps us understand the general shape of the polynomial's graph and its end behavior. Polynomials can have different degrees, and these degrees affect the graph and its end behavior significantly.

Leading Coefficient

The leading coefficient is the coefficient of the leading term in a polynomial. It is the number that multiplies the highest power of \(x\). In the polynomial:

• y = 3x^3 + 15x^2 - 6x + 5

The leading coefficient is \(3\). Knowing the leading coefficient is crucial because its sign (positive or negative) influences the direction in which the polynomial's ends will go. Positive leading coefficients generally indicate that the polynomial will rise to infinity as \(x\) becomes very large or very small, while negative coefficients indicate it will fall to negative infinity.

End Behavior Analysis

End behavior describes how the values of \(y\) change as \(x\) approaches infinity (\(\rightarrow +\infty\)) or negative infinity (\(\rightarrow -\infty\)). To determine the end behavior, we look at both the degree and the leading coefficient of the polynomial. For a polynomial like:

• y = 3x^3 + 15x^2 - 6x + 5

With an odd degree (\(3\)) and a positive leading coefficient (\(3\)), the end behavior follows these patterns:

• As \(x \rightarrow -\infty\), \(y \rightarrow -\text{infty}\)
• As \(x \rightarrow +\text{infty}\), \(y \rightarrow +\text{infty}\)

This means the left end of the graph goes down and the right end goes up. Analyzing end behavior helps us visualize the polynomial and predict its long-term trends. This scenario aligns with option C from the exercise (\(-\text{infty}\) for \(x < 0\) and \(+\text{infty}\) for \(x > 0\)).