Problem 15 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-x^{3}$$

Step-by-Step Solution

Answer

Plot the key points, draw a smooth S-curve through the origin extending top left to bottom right.

Step 1: Understand the Function Form

The function given is a cubic function, specifically in the form \( f(x) = -x^3 \). This means that it is an odd degree polynomial, which generally has an S-shaped curve. The negative sign indicates that the graph is reflected across the x-axis.

Step 2: Identify Key Features

Since \( f(x) = -x^3 \) is a simple cubic function, it has no stretches or shifts. Its key features are its roots, which is only at \( x = 0 \). The function passes through the origin (0,0), and because of the odd power, the end behavior is as follows: as \( x \to \infty, f(x) \to -\infty \) and as \( x \to -\infty, f(x) \to \infty \).

Step 3: Calculate Important Points

Since the function is symmetric about the origin, key points include \((0, 0)\). To find other points, choose values for \(x\), compute \(f(x)\), and plot: for example, when \(x = 1, f(x) = -1\); when \(x = -1, f(x) = 1\).

Step 4: Plot the Graph

Start plotting the points calculated: \((0,0), (1,-1), (-1,1)\). These indicate the direction in which the graph curves. Continue plotting a few more points like \((2, -8), (-2, 8)\) for a clearer shape. Draw a smooth curve through these points, extending it to show the end behavior.

Step 5: Verify the Graph

Check the graph to ensure it passes through the origin and follows the correct end behavior as described. The curve should extend from the top left to bottom right, showing a reflected cubic form.

Key Concepts

Polynomial Functions

Polynomial functions are a central concept in algebra and involve expressions that consist of variables raised to whole number exponents. They can have one or more terms, but the function's overall degree is determined by the highest exponent present. In the given exercise,

• the polynomial function is a cubic function, as indicated by the highest power being three.
• Cubic functions are interesting because they form an S-shaped curve on a graph.
• These functions can have various configurations, but the simplest form is when no additional terms stretch, shift, or modify the curve.

For example, in the function \( f(x) = -x^3 \), there aren't other polynomial terms nor additional constants affecting its graph beyond its cubic shape and reflection.
Polynomial functions often help us understand more complex equations, offering a basis that includes key real-world phenomena like acceleration in physics or population growth modeling.

Graph Reflection

Graph reflection involves flipping a graph over a specific axis, often resulting in a mirrored effect. This process is crucial in many mathematical functions, especially in understanding the behavior of negative signs in equations. In this specific exercise,

• the function \( f(x) = -x^3 \) indicates a reflection across the x-axis.
• This reflection occurs because of the negative sign in front of \( x^3 \).
• When you plot this function, you'll notice that it resembles a typical cubic graph but inverted.

Reflection modifies the end behavior of the graph. For instance, normally a cubic function graph for positive \( x^3 \) starts in the bottom left and moves to the top right. However, with reflection, it flips direction and starts in the top left and moves to bottom right. This transformation is essential for determining how the graph behaves as inputs become infinitely large in either positive or negative directions.

Odd Degree Polynomial

Odd degree polynomials, like cubic functions (degree three), possess unique characteristics compared to their even-degree counterparts. Here are some key attributes:

• They have specific end behaviors: for very large positive or negative inputs, the function tends to either positive or negative infinity.
• Typically, they feature a point where the graph crosses the x-axis.
• In this specific case of \( f(x) = -x^3 \), as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to \infty \).

The crossing behavior is due to their unpaired exponents, contributing to a natural S-shaped or zigzag path depending on the function's degree.
This crossing through the origin, as seen with the function's zero root at \( (0,0) \), highlights the distinct characteristics of odd degree polynomials that make them crucial for understanding polynomial graphs' intricate nature.