Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\sqrt{6} x^{6}-x^{5}+2 x-2$$

Step-by-Step Solution

Answer

As \(x\) approaches both \(\infty\) and \(-\infty\), \(P(x)\) approaches \(+\infty\).

Step 1: Identify the Leading Term

The leading term of the polynomial is the term with the highest power of \(x\). In this function, \(P(x) = \sqrt{6} x^{6} - x^{5} + 2x - 2\), the leading term is \(\sqrt{6} x^{6}\). This term will dominate the behavior of the polynomial for very large or very small values of \(x\).

Step 2: Determine the Leading Coefficient Sign

The coefficient of the leading term is \(\sqrt{6}\), which is a positive number. This indicates that the leading term \(\sqrt{6} x^{6}\) is positive as \(x\) becomes very large (\(x \to \infty\)) and also positive as \(x\) becomes very negative (\(x \to -\infty\)).

Step 3: Analyze Even Degree Behavior

Since the leading term \(\sqrt{6} x^{6}\) has an even degree (6), both ends of the graph of the function will go in the same direction. Specifically, as \(x \to \pm\infty\), \(P(x)\) will tend towards \(+\infty\) because the leading coefficient is positive.

Step 4: Summarize the End Behavior

The end behavior of the graph is determined by the leading term \(\sqrt{6} x^{6}\). Since the degree is even and the leading coefficient is positive, the graph rises to \(+\infty\) as \(x\) approaches both positive and negative infinity. Thus, as \(x \to \infty\), \(P(x) \to +\infty\); as \(x \to -\infty\), \(P(x) \to +\infty\).

Key Concepts

Leading Term

When studying polynomials, the leading term is of utmost importance. It holds the key to understanding how the polynomial behaves. This term is identified as the one with the highest power of the variable. For example, in the polynomial, \[P(x) = \sqrt{6} x^{6} - x^{5} + 2x - 2\], the leading term is \(\sqrt{6}x^6\).

The leading term is crucial because it dictates the end behavior of the polynomial. Essentially, when \(x\) takes on very large or very small values, other terms become insignificant in comparison to the leading term. It overwhelmingly influences the growth direction of the function as \(x\) trends towards positive or negative infinity.

Leading Coefficient

The leading coefficient is another important aspect of polynomials. It is the coefficient in front of the leading term. In the polynomial \(P(x) = \sqrt{6} x^{6} - x^{5} + 2x - 2\), the leading coefficient is \(\sqrt{6}\).

This coefficient, being a positive number, greatly influences the overall direction of the polynomial's end behavior.

• If the leading coefficient is positive, the ends of the graph will rise upwards as \(x\) approaches positive or negative infinity.
• If it were negative, the graph would fall towards negative infinity at both ends.

The sign of this key number directly determines whether the graph of the polynomial opens upwards or downwards at extreme values of \(x\). "},{