Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=\frac{x^{5}}{10}-x^{4} $$

Step-by-Step Solution

Answer

The function's end behavior is \( x \to \infty \) implies \( f(x) \to \infty \) and \( x \to -\infty \) implies \( f(x) \to -\infty \).

Step 1: Determine the Leading Term of the Function

The function given is \( f(x) = \frac{x^5}{10} - x^4 \). To understand the end behavior, we should focus on the leading term, which is the term with the highest degree. Here, that term is \( \frac{x^5}{10} \). The degree of the function is 5, which is odd.

Step 2: Analyze the Leading Coefficient

The leading coefficient of the function is \( \frac{1}{10} \). Since this fraction is positive and the degree is odd, it suggests that as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 3: Create a Table of Values

A table can confirm this behavior:| \( x \) | \( f(x) \) ||---------|---------|| -100 | \( -1.01\times10^{10} \) || -10 | \( -11000 \) || 0 | 0 || 10 | 9000 || 100 | \( 9.9\times10^{8} \) |As \( x \) increases, \( f(x) \) becomes more positively large, and as \( x \) decreases, \( f(x) \) becomes more negatively large.

Key Concepts

Leading Term

In any polynomial function, the leading term plays a crucial role in determining its overall behavior, especially as values approach infinity or negative infinity. For the function \( f(x) = \frac{x^5}{10} - x^4 \), the leading term is \( \frac{x^5}{10} \). This term is the one with the highest power of \( x \), which, in this case, is 5. It essentially dictates the polynomial's end behavior, or how the graph of the function behaves as \( x \) becomes very large (positively or negatively).

The leading term provides insights into the function's growth, since higher degree terms grow faster than lower degree ones as \( x \to \pm\infty \). Therefore, regardless of the other terms in the polynomial, the leading term gives a reliable prediction of the polynomial function's end behavior.

Leading Coefficient

The leading coefficient is the factor multiplying the leading term, which significantly impacts the direction of the polynomial's end behavior. For \( f(x) = \frac{x^5}{10} - x^4 \), the leading coefficient is \( \frac{1}{10} \).

This coefficient is positive, suggesting that as \( x \to \infty \), the function \( f(x) \) will also tend to positive infinity, causing the graph to rise to the right. Conversely, since the degree of the polynomial is odd, it implies that as \( x \to -\infty \), \( f(x) \) will tend toward negative infinity, causing the graph to fall to the left.

In essence, the sign of the leading coefficient helps determine whether the ends of the graph open upwards or downwards as they stretch far along the x-axis. Positive coefficients lead to opposite directions of end behavior for odd-degree polynomials.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial, providing vital information about the function's end behavior and the number of roots the polynomial can have. In our example, \( f(x) = \frac{x^5}{10} - x^4 \), the degree is 5 since the term \( x^5 \) holds the highest power.

Understanding the degree is crucial because:

• A polynomial's degree indicates the number of roots or solutions it can potentially have (though some could be complex or repeated).
• The degree determines the end behavior of the function:
  • If the degree is odd, the function will have opposite end behaviors at either side of the graph.
  • Even degrees result in the same end behavior on both sides of the graph.

Hence, the odd degree in this polynomial hints that its ends will extend in opposite directions along the y-axis.

Table of Values

Creating a table of values helps visualize and confirm the expected end behavior of a polynomial function, serving as a useful complement to theoretical analysis. For \( f(x) = \frac{x^5}{10} - x^4 \), a table of values can demonstrate how the function behaves for specific \( x \) values.

Let's look at some sample points:

• At \( x = -100 \), \( f(x) \) predicts a large negative value \((-1.01 \times 10^{10})\).
• At \( x = -10 \), it outputs \(-11000\), another negative number.
• At \( x = 0 \), \( f(x) = 0 \), showing the function passes through the origin.
• At \( x = 10 \), \( f(x) = 9000 \), a fairly large positive value.
• At \( x = 100 \), the prediction is \(9.9 \times 10^8\), a large positive value again.

These values align with the theoretical expectation that \( f(x) \) should grow increasingly large and positive as \( x \to \infty \), and similarly large but negative as \( x \to -\infty \). Thus, the table of values supports the analysis based on leading term and coefficient.