Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=2 x^{3}-3 x^{2}+x-1$$

Step-by-Step Solution

Answer

The graph of function \(f(x) = 2x^{3} - 3x^{2} + x - 1\) will fall to the left and rise to the right.

Step 1: Identify the degree of the polynomial

The degree of the polynomial function \(f(x) = 2x^{3} - 3x^{2} + x - 1\) is 3, which is the highest power of x in the polynomial.

Step 2: Identify the leading coefficient

The leading coefficient of the polynomial function is 2, which is the coefficient of the term with the highest power of x.

Step 3: Apply the Leading Coefficient Test

As per the Leading Coefficient Test: \n1. If the degree of the polynomial is odd, and the leading coefficient is positive, the graph falls to the left and rises to the right. \n2. If the degree of the polynomial is odd, and the leading coefficient is negative, the graph rises to the left and falls to the right. \nHere, our polynomial has an odd degree (3) and a positive leading coefficient (2), hence the graph of the function will fall to the left (as x approaches negative infinity) and rise to the right (as x approaches positive infinity).

Key Concepts

Polynomial Function Behavior

Understanding the behavior of polynomial functions can seem daunting, but with certain rules, it becomes much simpler. One way to determine a polynomial's end behavior is through the Leading Coefficient Test. For instance, the polynomial function given in the exercise, f(x) = 2x^3 - 3x^2 + x - 1, exhibits specific behaviors at the ends of the graph. These behaviors are linked to the function's journey as x approaches positive and negative infinity.

Think of the graph as a journey: for x values heading towards negative infinity, if our polynomial's degree is odd and its leading coefficient is positive, we can predict that the graph will start high and end low, resembling a journey from a mountain peak down to a valley. Conversely, as x heads towards positive infinity, our polynomial's graph is expected to rise indefinitely, much like a plane taking off into the sky.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable x that appears in the expression. In the given example, f(x) = 2x^3 - 3x^2 + x - 1, the term with the highest power is 2x^3. Thus, we say the degree of this polynomial is 3.

• If a polynomial has an odd degree, its end behavior will differ at each end of the graph. Imagine the ends of the graph as the two opposite sides of a roller coaster track.
• If a polynomial has an even degree, both ends will either point up or down together, reminiscent of a parabola.

This characteristic helps us predict the shape of the graph before even plotting it on the coordinate plane. The degree arms you with knowledge about whether you're dealing with a gentle hill or a twisty roller coaster ride in the graph's contour.

Leading Coefficient

Leading coefficient is the term used to describe the coefficient of the first term when a polynomial is arranged in standard form (highest degree to lowest degree). This small yet mighty number plays a significant role in shaping the graph of the polynomial function. In our exercise, the leading coefficient is the number 2 that multiplies the x^3 term.

This coefficient influences the graph's steepness and direction. As seen with the Leading Coefficient Test, it determines the initial and final behavior of the graph; a positive leading coefficient suggests that the right end of the graph shoots up towards infinity, while a negative one would point it downwards into the abyss. It's akin to knowing whether your rocketship (the graph) is launching upwards or crash-landing from the get-go!