Problem 29 - College Algebra

Question

For the following exercises, determine the end behavior of the functions. $$ f(x)=x\left(x^{2}-2 x-8\right) $$

Step-by-Step Solution

Answer

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

Step 1: Expand the Function

First, let's expand the function. The given function is \( f(x) = x(x^2 - 2x - 8) \). Distribute \( x \) to each term inside the parenthesis:\[ f(x) = x^3 - 2x^2 - 8x \].

Step 2: Identify Leading Term

The leading term of a polynomial function determines its end behavior. In \( f(x) = x^3 - 2x^2 - 8x \), the leading term is \( x^3 \), because it has the highest power.

Step 3: Determine End Behavior from Leading Term

The degree of the leading term is 3, which is odd. Additionally, the coefficient of \( x^3 \) is positive. For odd-degree polynomials with positive coefficients, as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 4: Conclusion

Therefore, the end behavior of the function \( f(x) = x^3 - 2x^2 - 8x \) is as follows:- As \( x \to \infty \), \( f(x) \to \infty \).- As \( x \to -\infty \), \( f(x) \to -\infty \).

Key Concepts

Understanding Polynomial Expansion

Polynomial expansion is a technique used to rewrite a polynomial in a simplified form by eliminating parentheses. In the given function, \( f(x) = x(x^2 - 2x - 8) \), the function is initially written in a factored form which involves multiplication.
To expand the polynomial, we distribute the \( x \) that is outside the parentheses to each term inside the parentheses. This is done by multiplying:

• \( x \cdot x^2 = x^3 \)
• \( x \cdot (-2x) = -2x^2 \)
• \( x \cdot (-8) = -8x \)

After performing these multiplications, the expanded form of the function is \( f(x) = x^3 - 2x^2 - 8x \). Through polynomial expansion, we have simplified the polynomial to help identify key components, such as the leading term.

Leading Term Identification

In polynomial functions, the leading term is crucial because it has the highest power (exponent). It significantly influences how the function behaves, especially as \( x \) approaches positive or negative infinity.
In the expanded function \( f(x) = x^3 - 2x^2 - 8x \), the leading term is \( x^3 \). We identify it by looking for the term with the largest exponent.
This term, \( x^3 \), holds the key to determining the end behavior of the polynomial. It dictates the direction of the graph slopes as \( x \) moves towards infinity (\( \infty \)) and negative infinity (\( -\infty \)). Identifying the leading term equips us with the ability to predict how the function behaves at extreme values of \( x \).

Characteristics of Odd-Degree Polynomials

Odd-degree polynomials, like the one in our example \( f(x) = x^3 - 2x^2 - 8x \), have degrees that are odd numbers, in this case, 3. These polynomials have unique end behaviors that differentiate them from even-degree polynomials.
The determining factors for their end behavior include:

• The degree of the polynomial (is it odd?)
• The sign of the leading coefficient (is it positive or negative?)

With our leading term being \( x^3 \), the degree is odd, and the coefficient is positive. For odd-degree polynomials with a positive leading coefficient, as \( x \to \infty \), \( f(x) \to \infty \), meaning the graph rises to the right. Conversely, as \( x \to -\infty \), \( f(x) \to -\infty \), so the graph falls to the left. This distinct characteristic helps students quickly estimate the visual graph behavior of the function across its entire domain.