Problem 63 - Algebra 2

Question

Identify the period of each function. Then tell where two asymptotes occur for each function. $$ y=\tan 1.5 \theta $$

Step-by-Step Solution

Answer

The period of the function \(y=\tan 1.5\theta\) is \(\frac{2\pi}{3}\) and the two asymptotes occur at \(\theta = 0\) and \theta = \frac{2\pi}{3}.

Step 1: Find the Period

The period of a general tangent function \(y=\tan Bx\) is given by \(\pi/B\). Given the function is \(y=\tan 1.5\theta\), B in this case would be 1.5. Hence, the period would be \(\pi / 1.5 = \frac{2\pi}{3}\).

Step 2: Find the Asymptotes

For a general tangent function \(y=\tan Bx\), the asymptotes occur at \(x = \pi n/B\), where n is an integer (0, 1, 2, 3, ...). For the given function, substituting B as 1.5, the asymptotes would occur at \(\theta = \frac{2\pi}{3} n\). Take n = 0 for the first asymptote to get \(\theta = 0\) and n = 1 for the second asymptote to get \(\theta= \frac{2\pi}{3}\).

Key Concepts

Period of a Function

In the study of trigonometric functions, understanding the concept of "period" is crucial. The period of a function guarantees that the function will repeat its pattern after a certain interval. For tangent functions, this interval is evaluated using the formula \(\pi/B\), where \(B\) is a constant multiplier of the variable in the function expression. Let's break this down further.

In the function \(y = \tan(1.5\theta)\), \(B = 1.5\). By plugging this value into the formula for the period of a tangent function, we find:

• Period \(= \pi / 1.5\)
• Which simplifies to \(\frac{2\pi}{3}\)

This means the graph of \(y = \tan(1.5\theta)\) will repeat its behavior every \(\frac{2\pi}{3}\) units along the \(\theta\)-axis. Understanding this repetitive nature helps in predicting the function's behavior over any span.

Tangent Function

The tangent function, specifically \(y = \tan(\theta)\), is a fundamental trigonometric function distinct for its periodic properties and undefined points. Unlike sine and cosine, which oscillate between -1 and 1, the tangent function can take any real value. This is due to the nature of its definition in the context of a right triangle or the unit circle: it is the ratio of the sine of an angle to its cosine.

Key aspects of the tangent function include:

• It repeats every \(\pi\) radians, traditionally.
• It has vertical asymptotes where the cosine of the angle is zero.
• It increases or decreases infinitely between these asymptotes.

For the function \(y = \tan(1.5\theta)\), the multiplier 1.5 affects both the period and the location of asymptotes, altering when and where the function shoots to infinity. This alteration is crucial in any applied scenario, as it changes the timing and "steepness" of the tangent curve.

Asymptotes

In mathematical terms, asymptotes are lines that a graph approaches but never actually touches or crosses. For the tangent function, vertical asymptotes occur at regular intervals, where the function tends to \(\pm \infty\). These occur when the cosine in the ratio definition is zero, leading the tangent to be undefined.

To find asymptotes in any tangent function \(y = \tan(Bx)\), the positions are determined by \(x = \pi n / B\), where \(n\) is an integer. This formula applies directly to \(y = \tan(1.5\theta)\):

• Substitute \(B = 1.5\) to get \(\theta = \frac{2\pi}{3}n\).
• Choose \(n = 0\) for the first asymptote: \(\theta = 0\).
• Choose \(n = 1\) for the second asymptote: \(\theta = \frac{2\pi}{3}\).

These asymptotes define the boundaries of the periodic segments, where the tangent curve will rise towards positive infinity on one side and fall towards negative infinity on the other side, without ever crossing these vertical lines. Understanding where these occur is vital for sketching and interpreting the graph of the tangent function.