Problem 52 - Algebra and Trigonometry

Question

In Exercises 45–52, graph two periods of each function. $$ y=\left|\tan \frac{1}{2} x\right| $$

Step-by-Step Solution

Answer

The graph of the function \(y = |\tan(\frac{1}{2}x)|\) is that of the tangent function stretched horizontally by a factor of 2 and reflected across the x-axis whenever it is negative. Two periods of this function will occur over an interval from 0 to \(4\pi\).

Step 1: Understand the function

The function given is \(y = |\tan(\frac{1}{2}x)|\). This is the absolute value of the tangent function, with its input divided by 2. The tangent function normally has a period of \(\pi\). However, by dividing its input by 2, its period is now \(2\pi\). Thus, two periods of this function would occur over an interval of \(4\pi\). Because this is an absolute value function, it should only return positive values.

Step 2: Determine key points to plot

To plot this graph effectively, it can be helpful to determine key points that will exist in each period. As a tangent function, important points will include: where the function crosses the x-axis, which will be at every multiple of \(\pi\), and where the function approaches negative and positive infinity, which will occur at \(\pi/2 + n\pi\), where n is an integer. In the case of the absolute value function, points that would normally approach negative infinity will instead mirror across the x-axis to approach positive infinity.

Step 3: Graph the function

Finally, plot the points determined in the previous step and connect them to create a smooth curve, remembering to mirror the portions of the graph that would normally be negative. The interval to be graphed should be from 0 to \(4\pi\) to show two complete periods.

Key Concepts

Tangent Function

The tangent function, written as \( \tan(x) \), is one of the primary trigonometric functions. It is derived from the ratio of sine and cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Unlike sine and cosine, the tangent function is not continuous across all real numbers because there are points where it is undefined; these points occur at odd multiples of \( \frac{\pi}{2} \) where \( \cos(x) = 0 \).

Key characteristics of the tangent function include:

• The function's value is 0 whenever \( x = n\pi \) for any integer \( n \).
• It approaches positive or negative infinity as \( x \) approaches multiples of \( \frac{\pi}{2} \), i.e., \( \frac{\pi}{2} + n\pi \).
• It exhibits a repeating pattern, making it periodic within its defined intervals.

Understanding these properties helps in graphing and manipulating the function under transformations, such as scaling and reflection.

Periodicity

Periodicity refers to the repetitive nature of trigonometric functions. The period of a function is the length of the smallest interval over which the function repeats itself. For the tangent function \( \tan(x) \), the period is \( \pi \).

In the modified function \( \tan(\frac{1}{2}x) \), the input \( x \) is divided by 2. This change affects the periodicity, stretching the graph horizontally and therefore doubling the period to \( 2\pi \). This means that the function will repeat its pattern every \( 2\pi \) units instead of every \( \pi \).

• Generally, for \( \tan(kx) \), where \( k \) is a constant, the period becomes \( \frac{\pi}{|k|} \).

Recognizing how inputs are modified allows us to precisely graph the function over its domain.

Absolute Value Function

The absolute value function, written as \( |x| \), transforms any input to a non-negative number. This transformation essentially mirrors any portion of a graph that dips below the x-axis. In the equation \( y = |\tan(\frac{1}{2}x)| \), the absolute value ensures that the output is always positive.

For example, where \( \tan(\frac{1}{2}x) \) would normally produce negative values, \( |\tan(\frac{1}{2}x)| \) will reflect these values into positive ones, creating a "V" shape around the axis where the lines would normally descend below the x-axis. The transformation through absolute value can be visualized as folding the graph up towards the positive y-axis. This symmetry results in a graph that does not exhibit the original tangent's negative dips.

Key Points in Trigonometry

Trigonometry heavily relies on specific key points for function behavior analysis and graphing purposes:

• Zero crossing points: These are where the function intersects the x-axis. For tangent, this is every \( n\pi \), where \( n \) is an integer.
• Undefined points: Points where the tangent function is undefined include every \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer. At these points, the function tends to asymptotes.
• Function periodicity: Recognize the repeating nature of these points and intervals.

Understanding these key points aids in sketching the accuracy of trigonometric function graphs and predicting their behavior over assigned intervals. Using these points in plotting ensures precision in reproducing function characteristics and nuances.