Problem 95 - Algebra and Trigonometry

Question

In Exercises 94–97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = -2x^3 + 6x^2 + 3x - 1\) is a cubic curve. The end behavior of the polynomial is: as \(x -> +∞, f(x) -> -∞\) and as \(x -> -∞, f(x) -> +∞\).

Step 1: Identify the Graphing Utility

Determine the graphing utility of choice. There are several online graphing tools available, such as Desmos, GeoGebra, or the graphing calculator.

Step 2: Input the Function

Enter the polynomial function \(f(x) = -2x^3 + 6x^2 + 3x - 1\) into the graphing tool.

Step 3: Adjust Viewing Rectangle

To show end behavior, adjust the viewing box to a sufficiently large range. This helps in visualizing the whole shape of the curve.

Step 4: Analyze the Graph

View the graph of the polynomial. A cubic function has a curve shape, either 'n' shaped or 'u' shaped with either tails of the curve going upward or downwards. Analyze the general shape and important parts, such as the intercepts.

Step 5: Determine the End Behavior

Observe the ends of the graph. Since the coefficient of the highest degree term (i.e., -2x^3) is negative, as \(x\) goes to positive infinity, \(f(x)\) will go to negative infinity, and as \(x\) goes to negative infinity, \(f(x)\) goes to positive infinity. Thus, the end behavior can be written as: as \(x -> +∞, f(x) -> -∞\) and as \(x -> -∞, f(x) -> +∞\).

Key Concepts

End Behavior of Polynomials

Understanding the end behavior of polynomial functions is crucial for predicting how the function behaves as the input values become very large or very small. For the polynomial function
\(f(x) = -2x^3 + 6x^2 + 3x - 1\),
we look at the leading term, which is \( -2x^3 \). This term dictates the function's end behavior because as \( x \) tends towards infinity or negative infinity, the other terms become negligible in comparison. Since the coefficient of the leading term is negative, the function falls to negative infinity as \( x \) approaches positive infinity and rises to positive infinity as \( x \) approaches negative infinity.

This property tells us the 'tails' of the graph will behave opposite to one another: one tail will head down to the right (as \( x -> +∞, f(x) -> -∞ \) ) and the other will head up to the left (as \( x -> -∞, f(x) -> +∞ \) ). Visualizing this behavior is useful when graphing the polynomial and understanding its general shape.

Cubic Functions

Cubic functions, such as \( f(x) = -2x^3 + 6x^2 + 3x - 1 \), are a type of polynomial function where the highest degree is three. These functions are characterized by their distinctive 'S' shape, which can manifest as either an 'n' or a 'u' shape, depending on the sign of the leading coefficient.

Characteristics of Cubic Functions:

• One to three real roots (x-intercepts).
• At least one local maximum and one local minimum.
• An odd function if the function is symmetric about the origin (which is not always the case).
• End behavior that has the 'left tail' and the 'right tail' heading in opposite directions.

Cubic functions present challenges and opportunities in graphing because of their potential bends and turns. Using a graphing utility helps to accurately plot the points and view these characteristics. Observing the turning points and intercepts offers a deeper understanding of the function's real-world behavior.

Graphing Utility in Algebra

A graphing utility is an invaluable tool when it comes to algebra, especially for understanding complex functions like polynomials. It allows students and mathematicians to visualize equations that are otherwise difficult to comprehend through standard algebraic manipulation.

To graph the function \( f(x) = -2x^3 + 6x^2 + 3x - 1 \) using a graphing utility:

• Select an appropriate graphing tool (like Desmos, GeoGebra, or a graphing calculator).
• Enter the equation precisely to ensure the graph reflects the correct function.
• Adjust the viewing window to capture the important aspects of the function, such as intercepts and turns, as well as to observe the end behavior adequately.

Graphing utilities often come with features like zoom, trace, and sliders that allow users to investigate specific points and behaviors of functions. Embracing these tools can significantly aid in the understanding of the underlying algebra and when solving complex problems.