Problem 15 - Algebra 2

Question

Find the amplitude, if it exists, and period of each function. Then graph each function. \(y=2 \tan \theta\)

Step-by-Step Solution

Answer

The function has no amplitude and a period of \( \pi \).

Step 1: Understanding Amplitude

For a tangent function of the form \( y = a \tan \theta \), there is no amplitude. Amplitude is a concept relevant for sine and cosine functions because it represents the maximum distance from the centerline. The tangent function has asymptotes and extends infinitely.

Step 2: Calculating Period

The period of the tangent function, \( \tan \theta \), is \( \pi \). The formula to calculate the period for \( y = a \tan(b\theta) \) is \( \frac{\pi}{b} \), where \( b \) is the coefficient of \( \theta \). In our function, \( b = 1 \), so the period is \( \frac{\pi}{1} = \pi \).

Step 3: Summary of Function Characteristics

The function \( y = 2 \tan \theta \) has no amplitude due to the nature of the tangent function, and its period is \( \pi \).

Step 4: Graphing the Function

To graph the function \( y = 2 \tan \theta \), plot the basic tangential curve which repeats every \( \pi \) units along the x-axis, keeping in mind the vertical scaling factor of 2 that influences the steepness of the curve. The function has vertical asymptotes at \( \theta = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \), where the function is undefined.

Key Concepts

Amplitude of Trigonometric Functions

In trigonometric functions like sine and cosine, amplitude is a key concept. It refers to the maximum height from the centerline of the wave. However, for the tangent function, like in our exercise with the function \( y = 2 \tan \theta \), amplitude doesn't apply. This is because the tangent function does not oscillate between a maximum and minimum value. Instead, it extends infinitely in both vertical directions.
Therefore, when you encounter a tangent function in trigonometry, remember that it won't have an amplitude to measure, unlike the more familiar sine and cosine waves.

Periodicity

Periodicity in trigonometric functions is the length it takes for the function to complete one full cycle. For the tangent function \( y = 2 \tan \theta \), this cycle length is recognized as the period. Generally, the period of the basic tangent function \( \tan \theta \) is \( \pi \).
The period can be calculated using the formula \( \frac{\pi}{b} \), where \( b \) is the coefficient of \( \theta \). In \( y = 2 \tan \theta \), since \( b = 1 \), the period remains \( \pi \). Every \( \pi \) units along the x-axis, the tangent function repeats its pattern.

• Short period: Repeats more frequently.
• Long period: Repeats less frequently.

Knowing the period helps in predicting the behavior and graphing of the trigonometric function accurately.

Graphing Trigonometric Functions

Graphing trigonometric functions like \( y = 2 \tan \theta \) involves understanding certain key components of the function. First, be aware of how the period affects the repetition of the graph. For this function, the period is \( \pi \), meaning the pattern resets every \( \pi \) units.
Secondly, consider the vertical stretching factor, which is given by the coefficient 2 in this case. This multiplier influences how steep the tangential curves appear on the graph compared to the basic \( \tan \theta \) function.

• Vertical scaling factor: A number that affects how stretched or compressed the function graph appears vertically.

To graph the function, start plotting at zero, progressing along the x-axis, and watching for where the function is undefined due to vertical asymptotes.

Asymptotes in Trigonometric Functions

Asymptotes are lines that a graph approaches but never truly touches or crosses. In trigonometric functions like \( y = 2 \tan \theta \), asymptotes are crucial for mapping out where the function is undefined.
For \( y = 2 \tan \theta \), vertical asymptotes occur at regular intervals along the \(\theta\)-axis. These are the points where the function heads towards infinity and is undefined. Specifically, for tangent function \( \theta = -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \), and so forth represent these vertical asymptotes.

• Vertical asymptotes: Lines parallel to the y-axis where the function becomes undefined and shoots towards positive or negative infinity.

Recognizing vertical asymptotes in trigonometric functions helps prevent errors in graphing and ensures an accurate representation of the inherently infinite behavior of the tangent curve.