Problem 176 - Precalculus

Question

For the following exercises, determine the end behavior of the functions. $$f(x)=(2-x)^{7}$$

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 Type of Function

The function given is \( f(x) = (2 - x)^7 \), which is a polynomial function because it can be expanded into a sum of powers of \( x \). The highest power of \( x \) after expansion will be \( x^7 \), confirming it is a polynomial of degree 7.

Step 2: Analyze the Degree and Leading Coefficient

The degree of the polynomial is 7, which is odd. The leading term after expanding \( (2-x)^7 \) will be \( -x^7 \) because the negative sign in \( (2-x) \) will cause the leading coefficient to be negative when expanded.

Step 3: Determine the End Behavior from the Leading Term

For a polynomial \( ax^n \) of odd degree \( n \) with a leading coefficient \( a \):- If \( a > 0 \), as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).- If \( a < 0 \), as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).Since the leading term is \( -x^7 \), \( a = -1 \) is less than 0.

Step 4: Conclusion on End Behavior

Given the analysis, as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \). This is the end behavior of the function \( f(x) = (2-x)^7 \).

Key Concepts

Understanding Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of variables, typically written as a sum of terms. Each term is made up of a constant called the coefficient, multiplied by the variable (usually noted as \( x \)) raised to a power. These functions are widely used due to their simple structure and easy manipulability.
For example, a polynomial can take the form \( f(x) = 3x^4 + 2x^3 - x + 5 \). In this expression, the terms are \( 3x^4 \), \( 2x^3 \), \( -x \), and \( 5 \).

• The degree of the polynomial is given by the highest power to which \( x \) is raised.
• Polynomial functions can be classified based on their degrees: linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.

Understanding these basics helps in analyzing more complex aspects such as end behavior.

Grasping the Degree of a Polynomial

The degree of a polynomial is an important feature, indicating the highest power of the variable in the polynomial when written in standard form. It significantly impacts the shape and behavior of the polynomial's graph.
For instance, revisiting our function \( f(x)=(2-x)^7 \), upon expanding, the degree becomes 7 because the highest exponent of \( x \) is 7.

• A higher degree generally means a more complex curve with possibly more turns.
• Odd-degree polynomials tend to have different tail behaviors on both ends of the graph.

The degree tells us not just about the number of potential solutions (roots), but also the characteristics of the graphical representation.

The Role of the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power in a polynomial. It plays a crucial role in determining the end behavior of the polynomial function.
In \( f(x)=(2-x)^7 \), the leading term after expansion is \( -x^7 \), making the leading coefficient \(-1\).

• A positive leading coefficient means that as \( x \to \infty \), the function grows towards \( \infty \) if the degree is odd.
• A negative leading coefficient means the opposite: as \( x \to \infty \), \( f(x) \to -\infty \).

The sign of the leading coefficient, together with the degree, allows us to predict how the graph behaves at very large or small values of \( x \).

Performing End Behavior Analysis

End behavior analysis refers to understanding what happens to the polynomial function when \( x \) values are very large (positive) or very small (negative). This analysis is crucial as it gives insights into the 'long-term' behavior of the function.
For our example, \( f(x)=(2-x)^7 \):

• The polynomial is of odd degree (7).
• The leading coefficient is negative (-1).

These characteristics tell us that:
- As \( x \to \infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).
Thus, the graph falls to the right and rises to the left, meaning one end of the graph heads downwards while the other heads upwards as \( x \) moves towards extreme values.