Problem 29 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=\tan \frac{\pi}{4} x$$

Step-by-Step Solution

Answer

The period of the function is 4.

Step 1: Identify the Function Formula

The given function is \( y = \tan \left( \frac{\pi}{4} x \right) \), which is a tangent function with a modified period.

Step 2: Understand the Period of the Tangent Function

The standard period of the tangent function \( y = \tan(kx) \) is \( \frac{\pi}{|k|} \). In this problem, \( k \) is \( \frac{\pi}{4} \).

Step 3: Calculate the Period

Substituting \( k = \frac{\pi}{4} \) into the period formula gives us:\[\text{Period} = \frac{\pi}{\left| \frac{\pi}{4} \right|} = 4\]Thus, the period of the function is 4.

Step 4: Graph the Function

To graph \( y = \tan \left( \frac{\pi}{4} x \right) \), first note that the x-intercepts occur at 0, 4, 8, etc., and vertical asymptotes occur at 2, 6, 10, etc., repeating every 4 units. The graph repeats itself for every interval of 4 along the x-axis.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can seem challenging, but understanding their basic shapes and properties helps greatly. Let's start by looking at the steps involved. Trigonometric graphs come from sine, cosine, tangent and other similar functions. When graphing, you generally need to identify a few key elements:

• Period: This shows how often the pattern of the graph repeats itself.
• Amplitude: This measures the height of the peaks or depth of the troughs from the graph's centerline. For tangent functions, the concept of amplitude doesn't apply as they stretch infinitely.
• Key points: Intercepts, peaks, valleys, and asymptotes are important points that guide the shape of the graph. These points help in sketching an accurate graph.

To graph a trigonometric function, such as the tangent, you'll follow these key points to plot it manually or by using a graphing calculator. Remember that tangent is unique, featuring vertical asymptotes where the function is undefined.

Tangent Function

The tangent function, often denoted as \( y = \tan(x) \), is one of the fundamental functions in trigonometry. Unlike the sine and cosine functions, tangent has a specific set of characteristics. It is periodic, but with a period of \( \pi \) instead of \( 2\pi \). Here are some features of the tangent function:

• Period: The basic period is \( \pi \), which means the pattern of the graph repeats every \( \pi \) units along the x-axis.
• Range: The range is all real numbers (-∞ to ∞), reflecting that the tangent values can increase or decrease without bound.
• Vertical Asymptotes: These occur at regular intervals, specifically at odd multiples of \( \frac{\pi}{2} \), where the tangent function tends towards infinity.

The standard tangent curve passes through the origin (0,0) and intersects every other unit along the x-axis. As it moves between these intercepts, it crosses the center point with positive and negative slopes, creating its distinctive shape.

Modifying the Period

In the function \( y = \tan \left( \frac{\pi}{4} x \right) \), the period has been modified from the standard tangent period of \( \pi \). When handling trigonometric functions, modifying the period involves altering the input value into the function. Here's a breakdown:

• The process starts with the function formula \( y = \tan(kx) \), where \( k \) determines how much the period changes.
• The new period of the tangent function is calculated using the formula \( \frac{\pi}{|k|} \).

In this specific example, \( k = \frac{\pi}{4} \), leading to a new period of 4. This means the graph of the tangent function now repeats every 4 units instead of the standard \( \pi \). Adjusting the period affects where the vertical asymptotes and intercepts appear on the graph, changing the frequency of repetition along the x-axis.