Problem 14 - Contemporary Precalculus

Question

(a) Graph \(y=x^{3}-2 x^{2}+x-2\) in the standard window. (b) Use the trace feature to show that the portion of the graph with \(0 \leq x \leq 1.5\) is not actually horizontal. \([\) Hint: All the points on a horizontal segment must have the same \(y\) -coordinate (why?).] (c) Find a viewing window that clearly shows that the graph is not horizontal when \(0 \leq x \leq 1.5\)

Step-by-Step Solution

Answer

Answer: After graphing the function and analyzing the graph in the given interval, we can observe that the y-coordinates are not the same, indicating that the graph is not horizontal in the interval [0, 1.5].

Step 1: (Part a: Graph the equation)

To graph the function \(y=x^3-2x^2+x-2\), use a graphing calculator or an online graphing tool. If using a graphing calculator, input the equation and adjust the viewing window to a standard setting (usually -10 to 10 for both x and y axis).

Step 2: (Part b: Analyze the graph between 0 and 1.5)

On the graph you made, zoom in around the interval \(0 \leq x \leq 1.5\). You can use the trace feature on your calculator to observe the movement of the curve in this specific interval. Pay close attention to the y-coordinates and observe if they are the same or not. As a horizontal segment would have the same y-coordinates at each point, confirming that the y-coordinates are changing will show that the portion of the graph is not horizontal.

Step 3: (Part c: Find a suitable viewing window)

To find a viewing window that clearly shows the graph is not horizontal in the given interval, adjust the window settings carefully. Use a smaller range for the x-axis, such as 0 to 1.5, and contextualize an appropriate range for the y-axis according to the graph, keeping in mind to capture the details of the curve. By observing the graph in this new window, it will become evident that the graph in the interval is not horizontal.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that simplifies the process of visualizing mathematical equations, particularly polynomial functions. It allows you to input polynomial equations, like the one in our exercise, and graph them instantaneously. This helps you see where curves rise or fall, where they intersect axes, and any points of interest such as maxima, minima, or inflection points.

• Begin by turning on the calculator and selecting the option to graph an equation.
• Enter the equation as it is; ensure correct input of the polynomial coefficients.

Still, the real advantage of using a graphing calculator lies in its features beyond basic graph plotting. Many calculators provide a trace function, zoom capabilities, and the ability to adjust window settings, making them invaluable for deeper analysis of polynomial curves.

Polynomial Curve Analysis

Analyzing a polynomial curve involves examining its shape and the changes in direction represented by the polynomial equation. The equation given, \(y = x^3 - 2x^2 + x - 2\), is a cubic polynomial, meaning its highest degree is 3.

• Such polynomials generally feature one or more critical points where they change from increasing to decreasing or vice versa.
• They can have inflection points where the concavity changes.

To analyze this specific polynomial, pay special attention to the sections where the curve seems flat or where changes in slope or curvature occur. For example, a common mistake might be perceiving parts of the curve as horizontal when they are not, which can be clarified by closely examining the increments in \(y\) over small intervals of \(x\).

Viewing Window Adjustment

Adjusting the viewing window on a graphing calculator might sound like a small detail, but it can make a significant difference in how we interpret a graph. The goal is to set the window so that it best captures relevant features of the polynomial.

• Start with a standard window, usually spanning from -10 to 10 for both axes.
• Narrow down the x-axis and y-axis limits to zoom into specific areas of interest.

For this exercise, to disprove the notion that the polynomial graph is horizontal between \(0 \leq x \leq 1.5\), adjust the window to focus on this range. Set the x-axis to, say, \(0\) to \(1.5\) and find a suitable small range on the y-axis to see if there is any variation. This closer inspection will provide more clarity and correct any oversight caused by inappropriate scale ranges.

Function Behavior Analysis

Analyzing the behavior of a function involves understanding not just the graphic output but also the mathematical implications behind the shapes. Different polynomial terms affect the behavior of \(y = x^3 - 2x^2 + x - 2\) as follows:

• The \(x^3\) term gives the curve an overall cubic nature, meaning it will tend to rise steeply at both ends.
• The \(-2x^2\) term influences the extent and location of the curve's peaks and troughs.
• Lower degree terms like \(x\) and constants slightly adjust the graph's path and position.

Function behavior analysis helps in predicting the graph's trajectory and pinpointing intervals where changes occur, such as between \(0 \leq x \leq 1.5\), confirming that the curve isn't horizontal, by identifying varying \(y\)-coordinates. A keen understanding of these influences increases students' ability to navigate polynomial functions effectively and understand their real-world applications.