Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a one-period interval. $$y=3 \tan \frac{1}{2} x$$

Step-by-Step Solution

Answer

Graph the function from \(0\) to \(2\pi\), showing zeros at \(0\) and \(2\pi\), and asymptote at \(\pi\).

Step 1: Understand the Tangent Function

The tangent function, represented as \(y = a \tan(bx)\), has a characteristic that it repeats itself every \(\pi\) for the basic function \(\tan(x)\). In the given function \(y = 3 \tan \frac{1}{2} x\), we identify \(a = 3\) and \(b = \frac{1}{2}\). Here, \(a\) represents the vertical stretch and \(b\) affects the period of the function.

Step 2: Determine the Period

The period of the tangent function \(\tan(bx)\) is given by \(\frac{\pi}{b}\). Substituting \(b = \frac{1}{2}\), we find the period: \[\text{Period} = \frac{\pi}{b} = \frac{\pi}{\frac{1}{2}} = 2\pi.\] So, the function \(y = 3 \tan \frac{1}{2} x\) completes one full cycle or period in the interval \([0, 2\pi)\).

Step 3: Identify Key Points within One Period

For \(y = \tan(x)\), key points within one period \([0, \pi)\) include the vertical asymptotes (where \(x = \frac{\pi}{2}, \frac{3\pi}{2}, ...\)), and the zeros (where \(x=0, \pi, 2\pi, ...\)). For \(y = 3 \tan \frac{1}{2} x\), we set \(\frac{1}{2}x = 0, \frac{\pi}{2}, \pi, ...\), leading to \(x = 0, \pi, 2\pi, ...\) for zeros, and vertical asymptotes at \(x = 2\pi, 4\pi, ...\). However, only \([0, 2\pi)\) is relevant for this graph.

Step 4: Sketch the Graph

Plot the tangent graph within the interval \([0, 2\pi)\). Mark zeros at \(x=0\) and \(x=2\pi\). Asymptotes appear at \(x=\pi\), and the graph approaches infinity near \(x=\pi\) without touching it. Due to the coefficient \(3\), the graph stretches vertically, making the slopes steeper compared to the standard tangent curve.

Key Concepts

Tangent Function

The tangent function is a trigonometric function often written as \(y = a \tan(bx)\). It is derived from the sine and cosine functions through the division \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). What makes the tangent function unique is its repeating pattern, known as periodicity, where it repeats every \(\pi\) radians for the standard function \(\tan(x)\). However, when modified by the parameters \(a\) and \(b\) in \(y = a \tan(bx)\), the graph can stretch or compress both vertically and horizontally.

• \(a\) affects the vertical stretch. In \(y = 3 \tan \frac{1}{2} x\), \(a = 3\), which makes the graph taller, making it look steeper. This vertical stretching does not affect the function's period but alters how quickly \(y\) values increase or decrease.
• \(b\) affects the horizontal stretch or compression, thereby changing the period. With \(b = \frac{1}{2}\), the function's period is expanded as it will complete one cycle in a longer interval compared to \(\tan(x)\).

Periodicity

Periodicity in trigonometric functions refers to how frequently a function repeats its pattern over the x-axis. For the tangent function \(y = \tan(bx)\), the period is determined by \(b\) and is represented by \(\frac{\pi}{b}\). This is because the standard tangent function \(\tan(x)\) completes one period over an interval of \(\pi\).

• In the function \(y = 3 \tan \frac{1}{2} x\), the period becomes \(2\pi\) since \(\text{Period} = \frac{\pi}{\frac{1}{2}} = 2\pi\). This means that instead of repeating every \(\pi\), it repeats after every \(2\pi\) interval.
• The extended period impacts how key features, such as asymptotes and zero crossings, are spread across the x-axis. In a \(2\pi\) cycle, you observe one complete tangent curve from start to finish.

Because of periodicity, knowing the period of a trigonometric function is crucial for graphing and analyzing its behavior accurately.

Graphing Functions

Graphing a function like \(y = 3 \tan \frac{1}{2} x\) involves identifying key properties influenced by constants \(a\) and \(b\). This helps trace the behavior over one complete cycle:

• **Zeros**: Occur when \(\sin(x) = 0\), which for \( \tan(x)\) typically occurs at \(x = 0, \pi, 2\pi...\), adjusted for \(b\), becoming \(0\) and \(2\pi\) in our function over one period. Zeros are where the graph crosses the x-axis.
• **Asymptotes**: These vertical lines where the tangent function approaches infinity but never actually crosses \(\frac{\pi}{2}+k\pi\). For \(y = 3 \tan \frac{1}{2} x\), they occur at \(x = \pi\) within one cycle, creating boundaries the function approaches.
• **Graph Shape**: The graph of \(y = 3 \tan \frac{1}{2} x\) is much steeper than \(y = \tan(x)\) due to the vertical stretch \(a = 3\). This means as you graph the function over \([0, 2\pi)\), you start and finish at the zeros, with a remarkable steep ascent and descent crossing an asymptote midway.

Graphing requires noting these points and drawing the distinct S-like shape of the tangent function, respecting its periodic rhythm and vertical transformations.