Problem 202 - Precalculus

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

Step 1: Identify the Leading Term

The leading term in a polynomial function is the term with the highest degree. For the given function \( f(x) = \frac{x^5}{10} - x^4 \), the leading term is \( \frac{x^5}{10} \). This term will dictate the end behavior of the function.

Step 2: Analyze the Degree and Coefficient

The degree of the leading term \( \frac{x^5}{10} \) is 5, which is odd. The coefficient is positive since \( \frac{1}{10} \) is positive. For functions where the degree is odd and the leading coefficient is positive, as \( x \to +\infty \), \( f(x) \to +\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 3: Construct a Table for End Behavior

Create a table with columns for \( x \), \( f(x) \), \( x \to +\infty \), and \( x \to -\infty \). Fill the table using the analysis from Step 2.| \( x \) | \( f(x) \) | Behavior ||---------|------------------|----------|| \( \to +\infty \) | \( \frac{x^5}{10} - x^4 \) | \( \to +\infty \) || \( \to -\infty \) | \( \frac{x^5}{10} - x^4 \) | \( \to -\infty \) |

Step 4: Conclusion

From the table and analysis, we conclude that as \( x \to +\infty \), \( f(x) \to +\infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \). This confirms the end behavior based on the degree and leading coefficient of the polynomial function.

Key Concepts

leading term

Understanding the leading term of a polynomial function is essential in determining the function's behavior, especially as the input values become very large or very small. In a polynomial function, the leading term is the term with the highest degree, which means it has the highest exponent of the variable. This term is the most significant in influencing the graph's shape as it extends towards infinity.For example, consider the polynomial function given in the exercise: \[ f(x) = \frac{x^5}{10} - x^4 \]The leading term here is \( \frac{x^5}{10} \). Even though there might be other terms like \(-x^4\), the term with the highest power of \( x \) (in this case, \( x^5 \)) will dominate the graph's behavior as \( x \) becomes very large, either positively or negatively. By focusing on the leading term, we simplify our work in predicting the end behavior of polynomial functions.

odd degree polynomial

Polynomials can be classified according to their degree, which is the highest exponent of the variable in the polynomial. An odd degree polynomial has an odd number as the highest exponent in its leading term. Odd degree polynomials have distinctive end behavior depending on their leading coefficient.For instance, consider the leading term \( \frac{x^5}{10} \) from our example. This term presents a degree of 5, which classifies it as an odd degree polynomial. Odd degree polynomials display distinct behaviors when considering their end points:

• If the leading coefficient (the numerical factor in the leading term) is positive, the function behaves such that as \( x \to +\infty \), \( f(x) \to +\infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).
• If the leading coefficient is negative, the behavior swaps: as \( x \to +\infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to +\infty \).

Since the polynomial in question has a positive leading coefficient (\( \frac{1}{10} \) is positive), it will tend towards positive infinity as \( x \) increases and negative infinity as \( x \) decreases. The degree of the polynomial thus plays a critical role in defining these behaviors.

polynomial end behavior analysis

Analyzing the end behavior of a polynomial involves looking at how the function behaves as its input approaches extreme values, both positively and negatively. This analysis is crucial because it gives insight into the overall direction of the function graph.In our exercise, the end behavior is primarily determined by the leading term \( \frac{x^5}{10} \), which is part of an odd degree polynomial. By knowing the degree and the sign of the leading coefficient, we can draw conclusions about the graph without needing to calculate specific values.

• The function \( f(x) = \frac{x^5}{10} - x^4 \) behaves such that as \( x \to +\infty \), \( f(x) \to +\infty \).
• As \( x \to -\infty \), \( f(x) \to -\infty \).

This analysis can be confirmed visually through a graph or systematically by observing changes in function values as \( x \) moves towards large positive and negative numbers. Graphically, understanding the end behavior helps in sketching the polynomial curve and anticipating how it behaves in natural and mathematical contexts.