Problem 126 - Precalculus Essentials

Question

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{3}-6 x^{2}+9 x+1$$

Step-by-Step Solution

Answer

This is a graph-based question. The accurate answer depends on the specific output of the graphing utility which can vary based on the tool used. Generally speaking, a basic cubic function like \(f(x)=x^{3}-6x^{2}+9x+1\) will feature a single curve that rises (increases), falls (decreases), then rises (increases) again, so the function is likely to be increasing for \(x < a\) and \(x > b\) and decreasing for \(a < x < b\) where a and b are turning points.

Step 1: Understand the Function

The function given is \(f(x)=x^{3}-6x^{2}+9x+1\). This is a cubic function, which typically has a curved graph. The graph might have one or more turning points, which is where the function changes from increasing to decreasing, or vice versa.

Step 2: Graph the Function

Graph the function using a graphing utility. Set the viewing rectangle values as mentioned - \(x\) ranged from -5 to 5 and \(y\) ranged from -5 to 5 each with increment of 1.

Step 3: Analyze the Graph

Once the graph is plotted, decide where it is increasing, decreasing, or constant. In general, the function is increasing when the slope of the graph is positive, decreasing when the slope is negative, and constant when the slope is zero.

Step 4: Find the Intervals

Lastly, mark the intervals on the \(x\)-axis where the function is increasing, decreasing, or constant. This can be done by visually inspecting where the curve of the graph is moving upwards (increasing), downwards (decreasing), or is flat (constant).

Key Concepts

Increasing and Decreasing Intervals

When analyzing cubic functions, understanding increasing and decreasing intervals is crucial. These intervals tell us how the function behaves as the input changes.
A function is said to be **increasing** on an interval if, as the input (\(x\)) increases, the output (\(f(x)\)) also increases. Mathematically, this can be represented by saying that for any two numbers \(x_1\) and \(x_2\) in the interval, if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\).
Conversely, a function is **decreasing** if, as \(x\) increases, \(f(x)\) decreases. In this case, for \(x_1 < x_2\), we have \(f(x_1) > f(x_2)\). The slope of the tangent line to the curve will be negative.
Identifying these intervals visually can be done by looking at the graph's slope:

• If the graph moves upward from left to right, it's increasing.
• If it moves downward, it's decreasing.
• If it is flat (a constant slope), then it is constant.

Cubic Functions

Cubic functions are polynomial functions of degree three. The general form of a cubic function is \(f(x) = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).
The graph of a cubic function is often characterized by its distinct shape, usually having one or two turning points, making it both interesting and challenging. A turning point is where the graph changes direction, which means a switch from increasing to decreasing, or vice versa.
Cubic functions can have:

• **One turning point**, leading to a simple S-curve, or
• **Two turning points**, making the curve more complex, often resembling a sideways S.

These shapes help in determining the function's increasing and decreasing intervals.
Understanding how these functions behave is essential for interpreting their graphs correctly.

Graphing Utility

Graphing utilities are technological tools, like calculators or computer software, used to plot functions and visualize their graphs. These utilities are invaluable for quickly graphing complex functions such as cubic functions.
When utilizing a graphing utility, ensure that:

• You correctly input the function expression. Misplaced parentheses or wrong coefficients can lead to incorrect graphs.
• The viewing window is appropriately set to capture all important features of the graph. For example, in this exercise, the \(x\) range \([-5, 5]\) gives a broad perspective on the function's behavior across its domain.
• You use the zoom or trace features to get a closer look at specific parts of the graph (such as turning points).

These tools allow for an easier analysis of the mathematical properties of the function, such as detection and verification of increasing and decreasing intervals and their points of inflection.