Problem 24 - College Algebra

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 'end behavior' of the graph of the polynomial function \(f(x) = 11x^4 -6x^2 + x + 3\) is such that as x approaches positive or negative infinity, the function f(x) approaches positive infinity.

Step 1: Identify Leading Coefficient and Degree

The highest power in the given polynomial function \(f(x) = 11x^4 -6x^2 + x + 3\) is 4; hence, the degree is 4. The coefficient of this term is 11, which is the leading coefficient. Because the leading coefficient, 11, is positive, and the degree, 4, is even, we can apply the rules of the LCT.

Step 2: Apply the Leading Coefficient Test

For a polynomial function with even degree and positive leading coefficient, the end behavior of the graph is as follows: As x approaches positive infinity (\(+\infty\)), y (the function value) also approaches positive infinity (\(+\infty\)) and as x approaches negative infinity (\(-\infty\)), y also approaches positive infinity (\(+\infty\)). In simpler terms, as x gets very large or very small, the function values will increase.

Step 3: Finalize the Solution

Therefore, the 'end behavior' of the graph of the given polynomial function \(f(x) = 11x^4 -6x^2 + x + 3\), using the Leading Coefficient Test, will be: as x approaches either positive infinity or negative infinity, the function f(x) will approach positive infinity.

Key Concepts

Understanding Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients. These functions are expressed in the form of \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n\) is a non-negative integer. Each part like \(a_ix^i\) is a term, and the degree of the polynomial is the highest power of \(x\) that has a non-zero coefficient.

In our example \(f(x) = 11x^4 - 6x^2 + x + 3\), the terms have powers 4, 2, 1, and 0 with respective coefficients 11, -6, 1, and 3. The degree of this polynomial is 4, since it is the highest exponent. Identifying the degree is crucial as it influences the shape and behavior of the graph of the polynomial function.

Exploring End Behavior

The 'end behavior' of a polynomial function describes how the function behaves as \(x\) approaches positive or negative infinity. This is important for understanding the overall direction of the graph as \(x\) moves far left or right on a coordinate plane.

Generally, the end behavior can be predicted using the Leading Coefficient Test (LCT), which considers the degree and leading coefficient of the polynomial. With this test:

• If the degree is even, the ends of the graph will go in the same direction.
• If the degree is odd, the ends will go in opposite directions.
• The sign of the leading coefficient will affect whether they rise or fall.

For instance, in our function \(f(x) = 11x^4 - 6x^2 + x + 3\), the degree is even (4), and the leading coefficient is positive (11), indicating that both ends tend to rise as \(x\) approaches \(\infty\) or \(-\infty\).

Characteristics of Even Degree Polynomials

Polynomials with an even degree, like \(f(x) = 11x^4 - 6x^2 + x + 3\), have distinct attributes that influence their graphical representation. Because the highest power is even, the graph will tend to mirror itself across the y-axis if plotted entirely.

Key characteristics of even degree polynomials include:

• Symmetrical behavior in terms of end directions—both ends of the graph of the function point up or down depending on the sign of the leading coefficient.
• Minimum or maximum points may appear in the "middle" of the graph, but at infinity, the behavior is consistent.

In our specific case, because the leading coefficient is positive and the degree is 4, both ends of the function tend to rise. Thus, the function approaches policy infinity as \(x\) goes to either plus or minus infinity.