Problem 24 - College Algebra

Question

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

Step-by-Step Solution

Answer

As \( x \rightarrow \, \infty \), \( f(x) \rightarrow \, -\infty \); as \( x \rightarrow \, -\infty \), \( f(x) \rightarrow \, \infty \).

Step 1: Identify the degree of the polynomial

The given function is \( f(x) = (2-x)^7 \). This expression is equivalent to \( -(x-2)^7 \), which is a single term polynomial when expanded. The degree of this polynomial is 7, which is odd.

Step 2: Determine the leading coefficient

When expanded, \( -(x-2)^7 \) has a leading term of \(-x^7\). Thus, the leading coefficient is -1, indicating it points in the opposite direction of a standard odd function.

Step 3: Apply end behavior rules for polynomials

For polynomials of degree \( n \):- If the degree \( n \) is odd and the leading coefficient is positive, the end behavior is: - As \( x \rightarrow \, \infty \), \( f(x) \rightarrow \, \infty \) - As \( x \rightarrow \, -\infty \), \( f(x) \rightarrow \, -\infty \) - If the degree \( n \) is odd and the leading coefficient is negative, like in our case with degree 7, the behavior is opposite: - As \( x \rightarrow \, \infty \), \( f(x) \rightarrow \, -\infty \) - As \( x \rightarrow \, -\infty \), \( f(x) \rightarrow \, \infty \)

Key Concepts

Polynomial Degree

Polynomials are mathematical expressions composed of variables raised to various powers, multiplied by coefficients, and added together. The degree of a polynomial is one of the most fundamental concepts in this context. This degree is determined by the highest power of the variable present in the polynomial expression. This signifies the overall 'order' of the polynomial.

In the given example, the expression \( f(x) = (2-x)^7 \) has been rewritten as \(-(x-2)^7\). Here, the polynomial is simple because it only contains one term after expansion. The highest exponent of \(x\) is 7, making it a seventh-degree polynomial.

This degree helps us understand the shape and structure of the polynomial's graph, influencing its end behavior significantly. Higher-degree polynomials tend to have more complex behaviors and shapes. Understanding the degree provides insight into how the function grows or decays as its input values become very large or very small.

Leading Coefficient

The leading coefficient is another key component of a polynomial's structure and behavior. It is the coefficient of the term with the highest degree in the polynomial. More simply, it's the number placed before the variable raised to the highest power.

For instance, when expanding \(-(x-2)^7\), we identify the leading term as \(-x^7\). The minus sign indicates a leading coefficient of \(-1\). This is the number that ultimately affects the direction in which the end behavior of the polynomial will head.

The size and sign of the leading coefficient help determine how rapidly a polynomial grows or shrinks and whether its graph flips upside down or remains upright. In our example, the negative leading coefficient \(-1\) means that the polynomial's ends will point in opposite directions compared to a similar polynomial with a positive leading coefficient.

End Behavior Rules

End behavior is a description of what happens to the values of a polynomial as the input \(x\) moves towards positive or negative infinity. It’s akin to predicting the long-term trends of the graph of a polynomial function.

The degree and leading coefficient of a polynomial largely dictate these patterns:

• For an odd degree polynomial with a positive leading coefficient (like \(x^3\)), as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \) and as \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \).
• Conversely, if the leading coefficient is negative (such as \(-x^7\)), the behavior is flipped. This means as \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \) and as \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \).

For our polynomial function \(-(x-2)^7\), the degree is odd, and the leading coefficient is negative. Thus, the end behavior is such that one end of the graph heads upwards while the other heads downwards, specifically:

• As \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \)
• As \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \)

These rules provide a quick and effective way to visualize the trends of polynomial graphs, significantly aiding in understanding their overall geometry.