Problem 86 - Precalculus

Question

What is the period of the function \(y=\tan \left(2 x+\frac{\pi}{2}\right) ?\) Use a graphing calculator to graph \(Y_{1}=\tan \left(2 x+\frac{\pi}{2}\right)\) \(Y_{2}=\tan \left(2 x-\frac{\pi}{2}\right),\) and \(Y_{3}=\tan \left(-2 x+\frac{\pi}{2}\right)\) in the same viewing window. Describe the relationships of \(Y_{1}\) and \(Y_{2}\) and \(Y_{2}\) and \(Y_{3}\)

Step-by-Step Solution

Answer

The period of \(y = \tan\left(2x + \frac{\pi}{2}\right)\) is \(\frac{\pi}{2}\). \(Y_2\) is \(\pi\) to the right of \(Y_1\), and \(Y_3\) is a reflection of \(Y_2\) over the y-axis with a \(\pi\) shift.

Step 1: Identify the Formula for the Period

The period of the tangent function \(y = \tan(ax + b)\) is given by the formula \(\frac{\pi}{|a|} \). Here, \(a = 2\), so the period is \(\frac{\pi}{2}\).

Step 2: Verify the Period with a Graphing Calculator

Enter the function \(Y_1 = \tan\left(2x + \frac{\pi}{2}\right)\) into the graphing calculator. Analyze how often the pattern repeats to confirm that the period is \(\frac{\pi}{2}\).

Step 3: Compare Functions Using a Graph

Graph \(Y_2 = \tan(2x - \frac{\pi}{2})\) and compare it to \(Y_1\). Note that \(Y_2\) is a horizontal translation of \(Y_1\) to the right by \(\pi\).

Step 4: Analyze the Relationship Between \(Y_2\) and \(Y_3\)

Graph \(Y_3 = \tan(-2x + \frac{\pi}{2})\). It is a reflection of \(Y_2\) over the y-axis, followed by a \(\pi\) horizontal translation.

Key Concepts

Period of Tangent Function

Understanding the period of a function is crucial when dealing with trigonometric functions like the tangent. The period of a function refers to the interval over which the function repeats its pattern. For the standard tangent function, the period is usually \(\pi\).
However, changes in the formula can affect the period. For the function \(y = \tan(2x + \frac{\pi}{2})\), we use the formula \(\frac{\pi}{|a|}\) to find the period, where \(a\) is the coefficient of \(x\).
In this case, \(a = 2\), leading to a period of \(\frac{\pi}{2}\).
This shorter period means the function completes a cycle more rapidly compared to the standard tangent function. Key Points to Note:

• The basic formula for determining the period of \(\tan(ax + b)\) is \(\frac{\pi}{|a|}\).
• As the value of \(a\) increases, the period decreases, causing the graph to repeat more frequently.

Horizontal Translation

In trigonometry, a horizontal translation shifts the entire graph of a function laterally, without altering its shape. Simply put, it means moving a graph left or right on the coordinate plane.
For the function \(Y_2 = \tan(2x - \frac{\pi}{2})\), it serves as a horizontally translated version of \(Y_1 = \tan(2x + \frac{\pi}{2})\).
The \(\pm \frac{\pi}{2}\) indicates how far and in which direction the curve is moved along the x-axis.

• A "+" sign before the constant results in a leftward shift.
• A "-" sign indicates a rightward displacement.

So for \(Y_2\), the graph is shifted to the right by \(\pi\). Understanding this is vital for predicting behavior and aligning graphs accurately when comparing functions.

Reflection Over the Y-axis

Another significant transformation in trigonometry is reflection, specifically over the y-axis. When a function is reflected over this axis, each point of the function is mirrored horizontally about the y-axis.In the problem provided, graph \(Y_3 = \tan(-2x + \frac{\pi}{2})\) comes from reflecting \(Y_2 = \tan(2x - \frac{\pi}{2})\).Key Reflection Insights:

• This transformation affects the sign of the coefficient of \(x\), changing from positive to negative or vice versa.
• The shape and behavior of the graph stay the same, but it is flipped horizontally.
• After reflection, a horizontal translation can also occur, as noted by the additional \(\pi\) shift in this case.

Understanding this process helps in visualizing changes in the graph's orientation and positioning.