Problem 50 - College Algebra

Question

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 \( f(x) \to -\infty \) as \( x \to -\infty \) and \( f(x) \to \infty \) as \( x \to \infty \).

Step 1: Identify End Behavior

Start by analyzing the leading term of the function to determine the end behavior. The given function is \[ f(x) = \frac{x^5}{10} - x^4. \]The leading term is \( \frac{x^5}{10} \). For a polynomial, as \( x \to \infty \), the end behavior is dominated by the leading term. Since the coefficient of \( x^5 \) is positive, \( f(x) \to \infty \). As \( x \to -\infty \), since 5 is an odd power, \( f(x) \to -\infty \).

Step 2: Create a Table of Values

Choose a set of x-values, both positive and negative large values, to observe the end behavior. Select x-values such as -100, -10, -1, 0, 1, 10, and 100.| x | f(x) ||---|---|| -100 | \( \frac{(-100)^5}{10} - (-100)^4 \) || -10 | \( \frac{(-10)^5}{10} - (-10)^4 \) || -1 | \( \frac{(-1)^5}{10} - (-1)^4 \) || 0 | \( \frac{0^5}{10} - 0^4 \) || 1 | \( \frac{1^5}{10} - 1^4 \) || 10 | \( \frac{10^5}{10} - 10^4 \) || 100 | \( \frac{100^5}{10} - 100^4 \) |

Step 3: Calculate Values

Calculate each value of \( f(x) \) using the expression:- \( x = -100 \): \( f(x) = \frac{(-100)^5}{10} - (-100)^4 = -1000000000.01 \)- \( x = -10 \): \( f(x) = \frac{(-10)^5}{10} - (-10)^4 = -11000 \)- \( x = -1 \): \( f(x) = \frac{(-1)^5}{10} - (-1)^4 = -1.1 \)- \( x = 0 \): \( f(x) = \frac{0^5}{10} - 0^4 = 0 \)- \( x = 1 \): \( f(x) = \frac{1^5}{10} - 1^4 = -0.9 \)- \( x = 10 \): \( f(x) = \frac{10^5}{10} - 10^4 = 9000 \)- \( x = 100 \): \( f(x) = \frac{100^5}{10} - 100^4 = 90000000 \)

Step 4: Confirm End Behavior

Based on the calculated values:- For large negative \( x \), such as \( x = -100 \), \( f(x) \) is a large negative value, confirming \( f(x) \to -\infty \) as \( x \to -\infty \).- For large positive \( x \), such as \( x = 100 \), \( f(x) \) is a large positive value, confirming \( f(x) \to \infty \) as \( x \to \infty \).Thus, the calculations are consistent with the analysis of the leading term.

Key Concepts

Leading Term in Polynomials

When working with polynomials like \[ f(x) = \frac{x^5}{10} - x^4 \],understanding the leading term is crucial. The leading term is the term with the highest power of \( x \).

• It is typically the first term you see when the polynomial is written in standard form (descending order of powers).
• In our example, \( \frac{x^5}{10} \) is the leading term because it contains \( x^5 \), the highest power of \( x \).

The leading term guides us in predicting the end behavior of the polynomial.It shows how the function behaves as \( x \) approaches positive or negative infinity. With a positive coefficient, it suggests that as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), the function behaves as \( f(x) \to -\infty \), given that the power is odd.

End Behavior Analysis

Analyzing the end behavior of polynomials involves looking at how the function acts as \( x \) moves towards very large positive or negative numbers.

• The end behavior is derived primarily from the leading term of the polynomial.
• In our polynomial function, \( f(x) = \frac{x^5}{10} - x^4 \), the dominant term is \( \frac{x^5}{10} \).

For this term:- As \( x \to \infty \): - Since the coefficient of \( x^5 \) is positive, \( f(x) \to \infty \).- As \( x \to -\infty \): - The fifth power (an odd exponent) implies \( f(x) \to -\infty \).This confirms that for very large values, the behavior aligns with a regular odd-degree function. Understanding this is essential for predicting how the graph of the function extends indefinitely.

Table of Values for Functions

Creating a table of values is an excellent way to visually affirm the end behavior of a function by selecting specific \( x \)-values and calculating \( f(x) \).

• Choose both very large and very small values of \( x \) to visualize the trend as \( x \) approaches infinity in either direction.

In our example, suggested \( x \)-values can be \(-100, -10, -1, 0, 1, 10,\) and \(100\):

• For \( x = -100 \): \( f(x) = -1000000000.01 \), a large negative number affirming \( f(x) \to -\infty \).
• For \( x = 100 \): \( f(x) = 90000000 \), a large positive number confirming \( f(x) \to \infty \).

This process helps confirm theoretical end behavior derived from the leading term by providing real numeric insights, supporting our initial analytic predictions.