Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to complete the table and graph the functions in the same viewing window. Make a conjecture about \(y_{1}\) and \(y_{2}\) $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\\\\hline y_{1} & & & & & & & \\\\\hline y_{2} & & & & & & & \\\\\hline\end{array}$$ $$y_{1}=\sec ^{4} x-\sec ^{2} x, \quad y_{2}=\tan ^{2} x+\tan ^{4} x$$

Step-by-Step Solution

Answer

It's not possible to report values or draw a graph using this format, but your graph and table would be complete upon following the steps above. Furthermore, your conjecture will be based on your graph and should spot any distinctive characteristic or relationship between \(y_{1}\) and \(y_{2}\).

Step 1 - Compute values

Using a graphing utility like a graphic calculator or an online software like Desmos, compute \(y_{1}\) and \(y_{2}\) for each given \(x\). Plug in each value of \(x\) into the equations \(y_{1}=\sec ^{4} x-\sec ^{2} x\) and \(y_{2}=\tan ^{2} x+\tan ^{4} x\) and compute the corresponding \(y_{1}\) and \(y_{2}\) values.

Step 2 - Fill into the table

Once you have computed the \(y_{1}\) and \(y_{2}\) values for each \(x\) given, fill the results into the provided table.

Step 3 - Graph the functions

Plot both functions \(y_{1}=\sec ^{4} x-\sec ^{2} x\) and \(y_{2}=\tan ^{2} x+\tan ^{4} x\) in the same graph. This can be done by inputting both formulas into the graphing utility.

Step 4 - Make a conjecture

Based on the graph, observe any patterns or relationships between \(y_{1}\) and \(y_{2}\). This could include overall shape, periodicity, symmetry, extrema or any other distinctive characteristics. Then, formulate a conjecture about the relationship between \(y_{1}\) and \(y_{2}\).

Key Concepts

Graphing Utilities

Graphing utilities are tools that help visualize mathematical functions, making them much easier to comprehend. For students and professionals alike, they are invaluable for understanding the behavior of functions in a visual format. Tools like graphing calculators, Desmos, or GeoGebra allow you to input mathematical expressions and immediately see their graphical representations.

With a graphing utility, you can manipulate variables and instantly observe changes, offering a dynamic learning experience. These utilities can handle complex functions, such as those involving trigonometric expressions like secant and tangent, making them essential for exploring trigonometric relationships.

While using these utilities, some tips include:

• Set appropriate viewing windows to ensure all critical parts of the graph are visible.
• Use trace features to find exact values of the functions at specific points.
• Zoom in and out to investigate different parts of the graph in more detail.

By mastering graphing utilities, you can effectively analyze and derive insights from various mathematical functions.

Secant Function

The secant function, often expressed as \(\sec(x)\), is one of the six standard trigonometric functions. It is the reciprocal of the cosine function, which means \(\sec(x) = \frac{1}{\cos(x)}\). Due to this relationship, the secant function inherits some characteristics from the cosine function.

Important features of the secant function include:

• Vertical asymptotes occur where \(\cos(x) = 0\), as division by zero is undefined.
• The function is periodic, with a period of \(2\pi\).
• Secant functions have no maximum or minimum values, as they can grow to infinity.

In the exercise, the function \(y_{1} = \sec^4{x} - \sec^2{x}\) emphasizes how transformations of basic trigonometric functions result in new behaviors and patterns. Elevating the secant function to even powers increases the steepness of its graph and affects its range.

Tangent Function

The tangent function, \(\tan(x)\), is another critical trigonometric function, defined as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This function is unique among the primary trigonometric functions due to its distinctive behavior and properties.

Key characteristics of the tangent function are:

• It has vertical asymptotes where \(\cos(x) = 0\), as these correspond to undefined points.
• Its period is \(\pi\), making it shorter than the period of the sine and cosine functions.
• The graph of the tangent function is continuous but not bounded, shooting to infinity as it approaches its asymptotes.

In the problem at hand, \(y_{2} = \tan^2{x} + \tan^4{x}\) illustrates how combining powers of the tangent function can alter its graph. Raising tan to higher powers accentuates its peaks and valleys, sometimes leading to more pronounced curves and sharper inclines.