Problem 9 - Algebra and Trigonometry

Question

\(5-10\) Use a graphing calculator or computer to decide which viewing rectangle \((a)-(\text { d) produces the most appropriate graph }\) of the equation. $$ \begin{array}{l}{y=10+25 x-x^{3}} \\ {\text { (a) }[-4,4] \text { by }[-4,4]} \\\ {\text { (b) }[-10,10] \text { by }[-10,10]} \\ {\text { (c) }[-20,20] \text { by }[-100,100]} \\ {\text { (d) }[-100,100] \text { by }[-200,200]}\end{array} $$

Step-by-Step Solution

Answer

Option (b) is the most appropriate graph view.

Step 1: Understand the function

The function given is a cubic polynomial, \( y = 10 + 25x - x^3 \). Cubic functions generally have at least one inflection point and can have turning points depending on the coefficients.

Step 2: Analyze the coefficients

The linear term (\(25x\)) and the cubic term (\(-x^3\)) influence the shape and direction of the curve. The cubic term will particularly dominate for large values of \(x\), causing the graph to eventually decrease as \(x\) increases past a certain point.

Step 3: Determine critical points analytically

To understand how the graph behaves, find its turning points by setting its first derivative equal to zero. The derivative is \(y' = 25 - 3x^2\). Set \(25 - 3x^2 = 0\) to find critical points, which leads to \(x^2 = \frac{25}{3}\) or \(x = \pm\sqrt{\frac{25}{3}}\).

Step 4: Evaluate extrema

The critical points are \(x = \pm\sqrt{\frac{25}{3}}\), approximately \(x = \pm2.89\). These are the potential turning points on the graph.

Step 5: Select suitable range

Based on the critical points, a viewing window that captures the behavior around these points while also showing a clear picture of the graph is needed. Option (a) \([-4,4] \text{ by } [-4,4]\) focuses on a small range, while (b) \([-10,10] \text{ by } [-10,10]\), (c) \([-20,20] \text{ by } [-100,100]\), and (d) \([-100,100] \text{ by } [-200,200]\) provide progressively larger views.

Step 6: Test viewing rectangles with graph

Test the function on a graphing calculator or computer using each provided view setting: (a) \([-4,4] \text{ by } [-4,4]\), (b) \([-10,10] \text{ by } [-10,10]\), (c) \([-20,20] \text{ by } [-100,100]\), and (d) \([-100,100] \text{ by } [-200,200]\). Analyze which one allows for a complete visualization of turning points and the general behavior of the function.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing complex functions like cubic polynomials. It allows you to input mathematical expressions and provides a graphical representation of those equations. This not only helps in understanding the shape and behavior of functions but also aids significantly in identifying critical features such as turning points. You can use a graphing calculator to experiment with different viewing windows, which are specific ranges for the x-axis and y-axis.

• Enter the function: Start by inputting the given cubic function, in this case, \( y = 10 + 25x - x^3 \), into the calculator.
• Adjust the viewing window: To properly visualize the function's critical features, adjust the x and y ranges on your graph. Trying options like \([-4,4]\) by \([-4,4]\) or \([-10,10]\) by \([-10,10]\) can be beneficial.
• Analyze: With the correct window settings, you can observe how the graph behaves, particularly around critical and turning points.

Using a graphing calculator, you can easily determine which viewing window provides the clearest picture of the function's behavior.

Critical Points

Critical points are where the function's derivative equals zero or where the derivative does not exist. For cubic functions like \( y = 10 + 25x - x^3 \), finding critical points is an essential step in understanding the function's behavior and shape. It helps identify where the graph has potential turning points.

• Calculate the derivative: For the given function, the derivative is \( y' = 25 - 3x^2 \).
• Set the derivative to zero: To find critical points, solve the equation \( 25 - 3x^2 = 0 \).
• Solve for x: This gives \( x = \pm\sqrt{\frac{25}{3}} \), which approximately equals \( x = \pm2.89 \).

These critical points suggest where the function might have local maximums or minimums, contributing crucially to understanding the cubic graph's turning points.

Turning Points

Turning points are specific places on the graph where the function changes direction. They occur at critical points where the derivative equals zero. For our function \( y = 10 + 25x - x^3 \), turning points are significant because they show where the graph can switch from increasing to decreasing or vice versa.

• Analyze the critical points: From the derivative \( y' = 25 - 3x^2 \), we found critical points at \( x = \pm\sqrt{\frac{25}{3}} \).
• Evaluate the behavior: At these points, the graph may have either a local maximum or minimum depending on the curvature.
• Graphically observe: Use your graphing calculator to confirm these turning points visually. They may appear as peaks or valleys on the graph.

Understanding turning points is crucial for a thorough analysis of cubic functions, as it reveals the nature and shape of the graph over its domain.