Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=-2 x^{4}-3 x^{2}+x-1 $$

Step-by-Step Solution

Answer

As \(x \to \infty\) or \(x \to -\infty\), \(f(x) \to -\infty\).

Step 1: Identify Leading Term

Determine the term with the highest exponent in the polynomial function, since this term will dominate the behavior as \(x\) approaches infinity or negative infinity. The leading term in \(f(x)=-2x^4-3x^2+x-1\) is \(-2x^4\).

Step 2: Determine Degree and Coefficient

The degree of the polynomial is 4, which is even, and the leading coefficient is \(-2\), which is negative. Both these characteristics influence the end behavior of the function.

Step 3: Determine End Behavior for Large Positive x

For very large positive values of \(x\), the term \(-2x^4\) will dominate the function. Since the coefficient is negative, \(-2x^4\) approaches negative infinity as \(x\) approaches positive infinity. Therefore, the function behaves as \(f(x) \to -\infty\) when \(x \to \infty\).

Step 4: Determine End Behavior for Large Negative x

For very large negative values of \(x\), the term \(-2x^4\) still dominates. When \(x\) is negative, raising it to an even power keeps the result positive, but the negative leading coefficient makes the result negative. Thus, \(-2x^4\) also approaches negative infinity as \(x\) approaches negative infinity. Thus, \(f(x) \to -\infty\) as \(x \to -\infty\).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomial functions take the general form:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0,\]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are constants (coefficients),
• \(x\) is the variable,
• \(n\) is a non-negative integer (degree of the polynomial).

The function's behavior, especially at extreme values of \(x\) (large or small), is driven by understanding these terms and how they work together. This is particularly important in determining end behavior, as you often see the variable raised to different powers.

Leading Term Analysis

The leading term of a polynomial is the term with the highest power of the variable. It determines the dominant behavior of the polynomial function as the variable grows extremely large or small. For example, in the polynomial \(f(x) = -2x^4 - 3x^2 + x - 1\), the leading term is \(-2x^4\).

• Why is the leading term important? Because it influences the shape and direction of the curve at its extremities.
• As \(x\) becomes very large in absolute value, the smaller terms become negligible, making the leading term crucial.

Analyzing the leading term helps to predict how the polynomial function behaves at the limits, aiding in a clearer visualization of the graph.

Degree and Coefficient

The degree of a polynomial is the highest power of the variable in the expression. In our example - \(f(x) = -2x^4 - 3x^2 + x - 1\) - the degree is 4, which is the exponent of the leading term \(-2x^4\).

• Degree Properties: The degree tells us how many times a polynomial can potentially intersect the x-axis and provides insight into its end behavior.

The leading coefficient is the constant multiplying the term with the highest degree, which is \(-2\) in this case.

• Leading Coefficient Influence: It affects the direction of the approach as \(x\) moves towards infinity or negative infinity.

Combining both, the degree and leading coefficient allow us to understand and predict the polynomial's end behavior effectively.

Positive and Negative Infinity

Understanding when a function approaches positive or negative infinity is crucial.As \(x\) moves towards positive infinity (to the far right on the x-axis) or negative infinity (to the far left), how the function behaves is primarily determined by its leading term's degree and coefficient.

• If the degree is even and the leading coefficient is positive, the function approaches positive infinity at both ends.
• If the degree is even and the leading coefficient is negative, the function approaches negative infinity at both ends, as with our example, \(-2x^4\).
• If the degree is odd, the ends will behave differently, which we do not cover in this specific case.

This concept helps in predicting how polynomial functions stretch or shrink and in which direction they turn as we evaluate larger values.