Problem 24 - Algebra and Trigonometry

Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. \(f(x)=-11 x^{4}-6 x^{2}+x+3\)

Step-by-Step Solution

Answer

The graph of the given polynomial function falls to the left and falls to the right.

Step 1: Identify the Degree of the Polynomial

The degree of the polynomial is the highest power that x is raised to in the polynomial. In the polynomial function \(f(x)=-11 x^{4}-6 x^{2}+x+3\), x is raised to the 4th power which is the highest out of all terms in the polynomial. Therefore, the degree of the polynomial is 4, which is an even number.

Step 2: Identify the Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In the given function, the term with the highest degree is \(-11x^{4}\). Hence, the leading coefficient is -11 which is negative.

Step 3: Use the Leading Coefficient Test

Once we have identified that the degree of the function is even and the leading coefficient is negative, we can use the Leading Coefficient Test which states: If the degree of the polynomial is even, and the leading coefficient is negative, then the graph falls to the left and falls to the right. Hence, this indicates the end behavior of the graph of the given function.

Key Concepts

Polynomial End Behavior

When we talk about the end behavior of a polynomial, it means how the function behaves as the input values become extremely large (positive infinity) or extremely small (negative infinity). Essentially, it describes the direction in which the tails of the graph of the polynomial go.
What happens as you move to the far left and far right on the graph can easily be predicted by looking at the polynomial's highest degree term and its coefficient.
The end behavior is significant in helping visualize how the polynomial’s graph will look, which assists in other areas of algebra such as graph sketching or solving inequalities.

Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In our example, with the polynomial function \(f(x)=-11x^{4}-6x^{2}+x+3\), the term \(-11x^{4}\) has the highest degree, which is 4.
This degree is important because it affects the shape and the end behavior of the graph. In general:

• Even degrees give the graph a symmetric, parabola-like shape.
• Odd degrees make the graph unsymmetric, resulting in one side of the graph pointing in a different direction.

Thus, knowing the degree allows us to start predicting how the graph might look and behave.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power in the polynomial. For our polynomial \(f(x)=-11x^{4}-6x^{2}+x+3\), the leading coefficient is \(-11\).
The leading coefficient, along with the degree, dictates the end behavior of the polynomial’s graph. If the leading coefficient is positive, the graph's long-term direction can vary based on the degree of the polynomial. Conversely, if it is negative, as in this case, it will significantly influence the direction in which the ends of the graph point.

Even Degree

An even degree, such as 4 in the polynomial \(f(x)=-11x^{4}-6x^{2}+x+3\), influences the general shape and symmetry of a graph. Polynomials with an even degree have graphs that are even, meaning they have a symmetrical shape like a bowl or an upside-down bowl, depending on the sign of the leading coefficient.
This symmetry implies that both ends of the graph either rise to positive infinity or fall to negative infinity. Understanding this helps to visualize potential graph shapes and behavior at infinity, which is crucial for various applications in math.

Negative Leading Coefficient

A negative leading coefficient, like \(-11\) in our polynomial, significantly impacts the direction of the graph's tails. If the degree is even, as here, both tails of the graph fall to negative infinity.
Imagine a regular parabola opening downwards. This is the end behavior expected because of the negative sign, making the graph go downwards on both sides, regardless of other terms in the polynomial. Recognizing this effect is especially useful when sketching graphs and predicting how polynomials behave with extreme values.