Problem 30 - Algebra 2

Question

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

Step-by-Step Solution

Answer

Amplitude does not exist. The period is \( \pi \).

Step 1: Write the function in standard form

First, we need to express the given equation in the standard form of a tangent function. The standard form of a tangent function is given by \( y = a \tan(b\theta + c) + d \). Let's rewrite the given equation \( 2y = \tan\theta \) as \( y = \frac{1}{2} \tan\theta \).

Step 2: Identify amplitude (if applicable)

In trigonometric functions, the amplitude is related to functions like sine and cosine and is represented by the coefficient \(a\) in the form \( y = a \sin(b\theta + c) + d \) or \( y = a \cos(b\theta + c) + d \). For the tangent function, amplitude is not defined because tangent has a range of all real numbers, so it does not have a maximum or minimum value.

Step 3: Determine the period

The period of a tangent function, in standard form \( y = a \tan(b\theta + c) + d \), is given by the formula \( \frac{\pi}{|b|} \). In our equation \( y = \frac{1}{2} \tan \theta \), since \( b = 1 \), the period is \( \frac{\pi}{1} = \pi \).

Step 4: Sketch the graph

The graph of \( y = \frac{1}{2}\tan \theta \) will look similar to the basic \( y = \tan\theta \) graph but vertically compressed by a factor of 2. The vertical asymptotes occur at \( \theta = \frac{\pi}{2} + k\pi \) for any integer \( k \), and the graph will repeat every \( \pi \) units.

Key Concepts

Understanding the Tangent Function

The tangent function, often represented as \( \tan \theta \), is one of the fundamental trigonometric functions. It's particularly unique compared to sine and cosine, primarily because it does not have a maximum or minimum value. Instead, the tangent function has an infinite range, meaning it can grow larger or smaller without bounds as the angle \( \theta \) changes.

• The standard form of a tangent function is \( y = a \tan(b\theta + c) + d \), where \( a \), \( b \), \( c \), and \( d \) are constants.
• In our problem, the function is simplified to \( y = \frac{1}{2} \tan \theta \), where \( a = \frac{1}{2} \), and \( b = 1 \).
• Unlike sine and cosine, the tangent function does not have an amplitude. This is because it does not oscillate between fixed values but instead increases or decreases without bound.

As \( \theta \) approaches certain angles, the value of \( \tan \theta \) approaches infinity or negative infinity, resulting in vertical asymptotes in its graph.

Exploring the Period of a Function

The period of a trigonometric function is the distance over which it repeats its values. Specifically for the tangent function, which we have in the standard form \( y = a \tan(b\theta + c) + d \), the period can be calculated using the formula \( \frac{\pi}{|b|} \).

• The constant \( b \) is crucial in determining how stretched or compressed the cycle of the tangent function appears.
• For the given function \( y = \frac{1}{2} \tan \theta \), the value of \( b = 1 \), which simplifies the period formula to \( \frac{\pi}{|1|} = \pi \).
• This period \( \pi \) indicates that the tangent function will repeat its pattern every \( \pi \) units along the horizontal axis.

Understanding the repeating nature of tangent functions is essential for identifying their characteristics when graphing.

Graphing Trigonometric Functions

Graphing the function \( y = \frac{1}{2} \tan \theta \) requires understanding how the tangent function behaves and its specific characteristics.

• The graph of the tangent function \( \tan \theta \) naturally forms a series of repeating curves, each separated by vertical asymptotes.
• To graph \( y = \frac{1}{2} \tan \theta \), note that the factor of \( \frac{1}{2} \) vertically compresses the graph, making it less steep than the standard tangent graph.
• Critical to graphing this function are the vertical asymptotes, which occur at \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is any integer.
• Each complete wave from one asymptote to the next replicates every \( \pi \) units, reinforcing the period concept described earlier.

When graphing, it's helpful to sketch light vertical lines at \( \frac{\pi}{2} + k\pi \) to mark the asymptotes, then draw the repeating tan curve between them. This visualization helps solidify understanding of the function's behavior across its domain.