Problem 46 - College Algebra

Question

Make a table to confirm the end behavior of the function. $$f(x)=-x^{3}$$

Step-by-Step Solution

Answer

The end behavior is confirmed: as \( x \to \\infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).

Step 1: Understand the Function

We begin by analyzing the given function, which is \( f(x) = -x^3 \). This function is a cubic function where the leading term is \(-x^3\), indicating it is a polynomial with a negative coefficient.

Step 2: Determine the End Behavior

The end behavior of a function is determined by the leading term. For \( f(x) = -x^3 \), as \( x \to \pm \infty \), the negative coefficient \(-1\) means the function will go to negative infinity. Specifically, as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).

Step 3: Select Test Values

To confirm this behavior, we can select specific test values for \( x \), such as \(-2, -1, 0, 1,\) and \(2\), and substitute them into the function to observe the output. This will help us see the trends in the values as \( x \) becomes larger positively and negatively.

Step 4: Create the Table

Create a table with the chosen \( x \) values and calculate the corresponding \( f(x) \) for each one:\[\begin{array}{c|c} x & f(x) = -x^3 \\hline-2 & -(-2)^3 = -(-8) = 8 \-1 & -(-1)^3 = -(-1) = 1 \0 & -(0)^3 = 0 \1 & -(1)^3 = -1 \2 & -(2)^3 = -8 \\end{array}\]

Step 5: Interpret the Table

From the table, observe that as \( x \) moves from negative to positive, \( f(x) \) decreases from \( 8 \) down to \(-8\). This confirms the end behavior: as \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to \infty \).

Key Concepts

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three. In general, a cubic function can be expressed in the form:

• \( f(x) = ax^3 + bx^2 + cx + d \)

Here, \( a, b, c, \) and \( d \) are constants, with \( a eq 0 \) determining the end behavior of the function. Since the degree is three, a cubic function might have up to three real roots and two turning points. These points occur where the function switches direction from increasing to decreasing or vice versa.
For example, in the function \( f(x) = -x^3 \), the coefficient \(-1\) in front of \( x^3 \) alters its shape, making it reflect across the x-axis compared to if the coefficient was \(+1\). Understanding these transformations helps in graphing and predicting the behavior of the function.

Polynomial Functions

Polynomial functions are a cornerstone in mathematics, defined by expressions consisting of variables raised to whole number powers and multiplied by coefficients. A general polynomial can be written as:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

Here, \( n \) is the highest power of \( x \), known as the degree, and \( a_n \) indicates the leading coefficient.
These functions can describe a wide array of real-world phenomena due to their ability to form curves, lines, and distinct behaviors given different coefficients and degrees.

• **Degree**: Determines the highest number of turns in the graph.
• **Leading Coefficient Sign**: Dictates the end behavior, setting the direction the ends of the graph face.

By analyzing these aspects, understanding the graph and roots becomes more accessible, which aids in solving polynomial equations like \( f(x) = -x^3 \).

Function Behavior Analysis

Analyzing the behavior of functions, particularly in terms of their long-term trends or end behavior, is an essential skill in calculus and algebra. The end behavior of polynomial functions, especially when looking at cubic functions like \( f(x) = -x^3 \), is predominantly influenced by their leading term. Here's what to look out for:

• **Positive Leading Coefficient and Odd Degree:** as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).
• **Negative Leading Coefficient and Odd Degree:** as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to \infty \).

The behavior at the ends of the graph tells us a lot about the nature of the polynomial. Using a table to calculate and visualize test values aids in interpreting the function's general direction. This analysis supports a deeper understanding of how different coefficients impact the overall graph.