Problem 24 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Leading Term

Since the function is given as \(f(x) = (2-x)^7\), we need to identify the leading term after the expression is expanded. The leading term of \((2-x)^7\) is \((-x)^7\) which simplifies to \(-x^7\).

Step 2: Determine the Degree and Leading Coefficient

The function \(f(x) = (2-x)^7\) can be rewritten as \((-x)^7 = -x^7\). Here, the degree of the polynomial is 7, and the leading coefficient is -1.

Step 3: Determine End Behavior Using Leading Term

The degree of the polynomial \(-x^7\) is odd, and the leading coefficient is negative. For a polynomial with an odd degree and a negative leading coefficient, as \(x \to -\infty\), \(f(x) \to +\infty\), and as \(x \to +\infty\), \(f(x) \to -\infty\).

Step 4: Conclude End Behavior

Therefore, the end behavior for \(f(x) = (2-x)^7\) is: as \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity; and as \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity.

Key Concepts

Polynomial Functions

Polynomial functions are a foundational concept in algebra and calculus. They are expressions composed of variables raised to whole-number exponents, and each term can have a coefficient. The general form of a polynomial function is:\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \] where \(a_n, a_{n-1},..., a_1, a_0\) are constants, and \(n\) is a non-negative integer.
Every polynomial function has several key characteristics, including its degree and the behavior as the input \(x\) becomes very large or very small, known as end behavior.

When analyzing polynomial functions like \((2-x)^7\), it's essential to understand how each term contributes to the function as a whole.

Leading Term

The leading term in a polynomial is the term with the highest power of \(x\) after expanding the expression.
It significantly influences the polynomial's end behavior.

In the function \(f(x) = (2-x)^7\), the leading term emerges after expansion as \((-x)^7\), which simplifies to \(-x^7\). - It involves both the highest exponent and the leading coefficient. - The leading term tells us about the rate and direction in which the function grows as \(x\) moves toward extreme positive or negative values. The leading term is crucial because it dictates the behavior of the entire polynomial for very large or very small \(x\). This is why it plays such a significant role in determining end behavior.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable \(x\) in its expression.
In our example, \(f(x) = (2-x)^7\), the degree becomes evident once we recognize the leading term \((-x)^7\).
The degree of this function is 7.

Understanding a polynomial's degree is important because:

• The degree indicates how many roots a polynomial can have.
• It helps in determining the shape and the number of turns in the graph of the polynomial function.
• Most critically, it is key in analyzing the end behavior of the function.

For an odd-degree polynomial like our example, the ends of the graph go off in opposite directions. This is why the end behavior involves \(\to \infty\) in opposite directions when \(x\) changes from \(-\infty\) to \(+\infty\).

Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest power (or degree).
In \(f(x) = (-x)^7 = -x^7\), the leading coefficient is \(-1\).
This negative value indicates that the graph of the polynomial will extend in opposite directions from what it would if the coefficient were positive.

Here’s why understanding the leading coefficient is essential:

• It affects the direction of the polynomial's end behavior. A negative leading coefficient flips the graph over the x-axis compared to a positive coefficient.
• When combined with the degree, it allows for a clear prediction of the polynomial's end behavior.

In our function’s context, the negative leading coefficient combined with the odd degree results in the graph moving toward positive infinity as \(x\) approaches negative infinity, and moving toward negative infinity as \(x\) approaches positive infinity.