Problem 50 - College Algebra

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

As \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 1: Identify the Leading Term

The leading term of the polynomial function will dictate the end behavior. In the function \( f(x) = \frac{x^5}{10} - x^4 \), the leading term is \( \frac{x^5}{10} \) because it has the highest power of \( x \).

Step 2: Determine End Behavior Based on Leading Term

Since the leading term is \( \frac{x^5}{10} \), a positive coefficient and an odd degree, the function will have an end behavior where as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 3: Create a Table for End Behavior

Create a table with two columns: \( x \) values and corresponding \( f(x) \) values. Choose \( x \) values approaching \( \infty \) and \( -\infty \).| \( x \) | \( f(x) \) ||-------------------|-----------------|| \( 1000 \) | \( f(x) > 0 \) || \( -1000 \) | \( f(x) < 0 \) |These chosen values reflect how \( f(x) \) behaves as \( x \) becomes very large or very small.

Step 4: Verify with a Graph (Optional)

To further confirm the end behavior, you can sketch the graph of \( f(x) \) or use graphing technology. Observing the graph should show that as \( x \to \infty \), \( f(x) \to \infty \); and as \( x \to -\infty \), \( f(x) \to -\infty \), confirming the deductions from the leading term analysis.

Key Concepts

Leading Term in Polynomial Expressions

In polynomial expressions, the leading term is crucial in determining the end behavior of the function. The leading term is the term with the highest degree, which means it has the highest power of the variable, typically represented as \( x \). Its coefficient and degree can tell us a lot about how the polynomial will behave for large values of \( x \).For example, in the polynomial function \( f(x) = \frac{x^5}{10} - x^4 \), the leading term is \( \frac{x^5}{10} \). This term is crucial because the power of \( 5 \) is the highest among all terms.

• If the degree of the leading term is odd and the leading coefficient is positive, as \( x \to \infty \), \( f(x) \to \infty \); and as \( x \to -\infty \), \( f(x) \to -\infty \).
• Conversely, if the degree were even, the end behavior would flip differently.

Understanding and identifying the leading term helps you predict and analyze the function's behavior without necessarily computing many individual values.

Graphing Polynomial Functions

Graphing polynomial functions is a tool we can use to visually confirm the behavior predicted by the leading term. Creating a graph of the polynomial helps to see how the function behaves as \( x \) moves towards large positive or negative values.Plotting the function \( f(x) = \frac{x^5}{10} - x^4 \) using graphing software or manually sketching can provide a clear visualization. Here’s what you typically look for:

• Slope at Extremes: Look at how the ends of the graph rise and fall. For \( \frac{x^5}{10} \), you’ll see the curve heads upward on the right and downward on the left.
• Intersections and Turning Points: While these are not always covered by the leading term, they can inform about where the graph might cross the axis or peak midway.

Graphing helps not only verify the end behavior but also offers insights into the overall shape of the polynomial, helping to clear up any uncertainty from a purely numerical approach.

Table of Function Values

Creating a table of function values is an approach that lays out how the function behaves over selective points, particularly useful in confirming the theoretical end behaviors determined by the leading term.For our example function \( f(x) = \frac{x^5}{10} - x^4 \), constructing a table involves:

• Choosing values for \( x \) far in the positive and negative directions. For instance, \( x = 1000 \) and \( x = -1000 \).
• Calculating \( f(x) \) for these \( x \) values to observe the output.

By doing this, it confirms that as \( x \) increases to a very large positive number, \( f(x) \) is positive, and as \( x \) moves towards a large negative number, \( f(x) \) becomes negative. This table effectively supplements the graph and leading term insights, providing concrete numerical evidence that can be compared against theoretical expectations.