Problem 24 - College Algebra

Question

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: Analyze the Leading Term

Consider the function \( f(x) = (2-x)^7 \). The leading term of this function when expanded would be \((-x)^7 = -x^7\) because the highest degree term in the polynomial dominates end behavior. It's important to note that \(x^7\) has an odd power of \(x\).

Step 2: Identify the End Behavior of \(x^7\)

The function \(x^7\) is a simple power function with an odd exponent. As \(x\) approaches positive infinity, \(x^7\) also approaches positive infinity, and as \(x\) approaches negative infinity, \(x^7\) approaches negative infinity.

Step 3: Determine the Impact of the Negative Sign

Since the leading term derived from \((-x)^7\) is \(-x^7\), this negative sign affects the end behavior. Specifically, it reverses the end behavior seen with \(x^7\). Thus, as \(x\) approaches positive infinity, \(-x^7\) approaches negative infinity. Conversely, as \(x\) approaches negative infinity, \(-x^7\) approaches positive infinity.

Step 4: Conclusion on End Behavior

Using the analysis in previous steps, the end behavior of \(f(x) = (2-x)^7\) can be summarized as: As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity. As \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity.

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression involving a sum of powers of a variable combined with coefficients. Functions like these are crucial in math because they can describe a wide variety of scenarios and relationships. In essence, a polynomial function takes the form:

• The general form is: \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)
• Here, \(a_i\) are coefficients, and \(n\) is a non-negative integer representing the highest degree.

Polynomials can be as simple as a constant or could feature complex combinations of terms. Analyzing the degree and leading coefficient of a polynomial helps in understanding its growth and behavior as the variable either shrinks or grows excessively. For the function in question, \(f(x) = (2-x)^7\), recognizing it as a polynomial of degree 7 provides insight into its long-term trends, which are driven by its highest power term.

Leading Term

The leading term of a polynomial function is the term with the highest power of the variable. In many ways, this term dictates the end behavior of the function because as the variable grows large (whether positive or negative), this term becomes overwhelmingly dominant over the others.
For example, if you have a polynomial like \(f(x) = -x^7 + lower~degree~terms\), its leading term is \(-x^7\). This term will control the polynomial's direction as \(x\) heads towards infinity in either direction.
The influence of the leading term becomes clearer when you recognize its basic pattern, which becomes the defining factor in graphing or predicting outcomes as \(x\) becomes substantially large or small. In cases with multiple variables, the leading term is assumed by rearranging the function to highlight the higher degree polynomial part through simplification or expansion. In conclusion, the leading term not only provides us with the degree but also a valuable piece of data to foresee the behavior at extremities.

Odd Exponent

Exponents play a crucial role in determining the overall behavior and characteristics of polynomial functions. An odd exponent is associated with a power term like \(x^7\), and it implies a certain symmetry around the origin for basic power functions like \(x^n\).
Odd exponents have characteristics that are quite predictable.

• As the variable \(x\) becomes very large positively, any term like \(x^7\) becomes very large positively as well.
• Conversely, as \(x\) moves negatively towards minus infinity, \(x^7\) swings into deep negative values.

This symmetry about the origin is what makes odd exponents unique within polynomial structures, as opposed to even exponents, which create a mirrored effect about the y-axis. This feature of odd exponents is apparent in the polynomial function \(f(x) = (2-x)^7\), where the leading term derives the odd exponent of 7 from its simplified form, influencing how the function behaves at its asymptotes or infinity points.

Negative Sign Influence

The negative sign in front of a leading term can switch the direction of the polynomial function's end behavior. For instance, examine \(-x^7\); the negative sign before the \(x^7\) causes an inversion.
Without a negative, when \(x\) approaches infinity, \(x^7\) rises to positive infinity. However, \(-x^7\) plunges to negative infinity when \(x\) swells positive.

• Similarly, when \(x\) sinks to negative infinity, \(-x^7\) ascends to positive infinity, reversing \(x^7\)'s natural negative dip.
• This inversion impacts the entire function so that its graph flips, reflecting the opposite behavior across quadrants impacted by its leading coefficient's sign.

Understanding this reversal is pivotal for mastering end behavior predictions of polynomial functions affected by a negative sign. Such comprehension ensures accurate graphing and anticipation of a function's behavior across infinite extents, as seen in \(f(x) = (2-x)^7\), where the negative leading term alters expectations otherwise set by an odd-powered polynomial.