Problem 18 - College Algebra with Corequisite Support

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=x^{3} $$

Step-by-Step Solution

Answer

As \( x \to + fty \), \( f(x) \to + fty \); as \( x \to - fty \), \( f(x) \to - fty \).

Step 1: Identify the Leading Term

The function given is \( f(x) = x^3 \). For polynomials, the end behavior is determined by the leading term, which is the term with the highest degree. Here, the leading term is \( x^3 \).

Step 2: Determine the Degree of the Leading Term

Since \( x^3 \) is the leading term, the degree of this polynomial is 3. This degree is odd, which will impact the end behavior of the function.

Step 3: Determine the Leading Coefficient

The leading coefficient is the coefficient of the leading term \( x^3 \). In this case, the leading coefficient is 1.

Step 4: Analyze the End Behavior Based on Degree and Leading Coefficient

For polynomials with an odd degree and a positive leading coefficient, like \( x^3 \), as \( x \to +fty \), \( f(x) \to +fty \), and as \( x \to -fty \), \( f(x) \to -fty \).

Step 5: Conclude Function’s End Behavior

The end behavior of the function \( f(x) = x^3 \) is such that as \( x \) approaches positive infinity, \( f(x) \) approaches positive infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.

Key Concepts

Degree of Polynomial

The degree of a polynomial is essentially the highest power of the variable in that polynomial. It dictates many aspects of the polynomial's behavior and shapes the graph of the function. For a given polynomial function, you always identify the degree by looking for the term with the largest exponent.

For instance, in the function \( f(x) = x^3 \), you see that the highest power of the variable \( x \) is 3, which makes the degree of this polynomial 3. This tells us not just about how the function behaves as \( x \) becomes very large or very small, but also how many times the function might change direction (zigzag) if we were to graph it. Understanding the degree is crucial in predicting how a polynomial function behaves towards both positive and negative infinity.

Leading Coefficient

The leading coefficient of a polynomial plays a vital role in determining the polynomial's end behavior. It is the coefficient of the term with the highest degree in the polynomial, usually found in front of the variable with the largest exponent.

In the function \( f(x) = x^3 \), the leading coefficient is 1. This number can significantly influence the direction of the graph as \( x \) approaches infinity or negative infinity.

When the degree is odd and the leading coefficient is positive, as in this function, the end behavior will show the graph rising to positive infinity on one end and falling to negative infinity on the other. Conversely, if the leading coefficient was negative, the end behavior would reverse—a crucial distinction for understanding how the graph will look at extreme values of \( x \).

• Positive leading coefficient: graph rises to the right
• Negative leading coefficient: graph falls to the right

Odd Degree Polynomials

Odd degree polynomials, like \( f(x) = x^3 \), have certain characteristics in terms of their behavior and symmetry. An odd degree signifies that the polynomial can have a certain asymmetrical behavior when graphed.

Basically, for an odd degree polynomial, the ends of the graph will go in opposite directions. In the function \( f(x) = x^3 \), as the degree is odd (3, in this case), you can predict that one side of the graph will head upwards as \( x \) increases, while the other side will go downward as \( x \) decreases.

This opposing directionality is a hallmark of odd degree polynomials, contrasting with the same-direction behavior of even degree polynomials. This clue helps in sketching or understanding the graph of such functions without detailed plotting. Remember, the crucial aspect here is asymmetry: one end heads to infinity while the other towards negative infinity.

Asymptotic Behavior

Asymptotic behavior is all about understanding how the graph of a function behaves at extreme values. For polynomials, this involves what happens as \( x \) approaches positive infinity or negative infinity, known as the end behavior.

In the function \( f(x) = x^3 \), which we have established has an odd degree and a positive leading coefficient, the asymptotic behavior is predictable: as \( x \to +\infty \), \( f(x) \to +\infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

• Positive \( x \) leads to positive \( f(x) \)
• Negative \( x \) leads to negative \( f(x) \)

Recognizing this pattern can save time and effort in foreseeing how a polynomial function's graph stretches and shrinks. Asymptotic analysis provides a "big picture" look, especially useful when considering large values of \( x \).