Problem 31 - Algebra 2

Question

a. Open-Ended Write a tangent function. b. Graph the function on the interval \(-2 \pi\) to 2\(\pi .\) c. Identify the period and the asymptotes of the function.

Step-by-Step Solution

Answer

The tangent function \(y = \tan(x)\) has a period of \(\pi\) and asymptotes at \(x = n\pi + \frac{\pi}{2}\), for any integer n, when graphed over the interval \(-2\pi\) to \(2\pi\).

Step 1: Define Tangent Function

A simple tangent function to define and use for this analysis could be \(y = \tan(x)\). This function will be graphed and analyzed for its period and asymptotes.

Step 2: Graph the Function

The above function is graphed on the interval \(-2\pi\) to \(2\pi\) using graphing software or a graphing calculator. Take note of the curve's shape and the points at which it seems to 'break off'.

Step 3: Identify the Period

The period of a tangent function is \(\pi\) units. This is the distance between one peak or valley of the graph to the next corresponding peak or valley.

Step 4: Identify the Asymptotes

Asymptotes for a tangent function can be found at \(x = n\pi + \frac{\pi}{2}\), for any integer n. These asymptotes are vertical lines showing where the function approaches positive or negative infinity.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions like the tangent function can be an insightful way to understand their properties and behavior. For the function \(y = \tan(x)\), the graph depicts a repeating pattern often marked by abrupt breaks or vertical asymptotes. When plotting \(y = \tan(x)\), you'll notice the function climbs from negative infinity to positive infinity, then resets and repeats this pattern. It's helpful to plot this graph in stages:

• Choose an appropriate interval. Here, it's from \(-2\pi\) to \(2\pi\).
• Identify key values such as \(-\pi, 0, +\pi\), and within these, observe how the graph behaves near the vertical asymptotes.
• Note that the graph crosses the x-axis at multiples of \(\pi\).

Visualizing this graph will help cement your understanding of how the tangent function behaves within a complete period.

Period of Trigonometric Functions

The period of a trigonometric function is essentially the length of one full cycle of the wave pattern on a graph. For the tangent function \(y = \tan(x)\), this period is \(\pi\). This is distinct from sine and cosine functions that have a period of \(2\pi\).
To determine the period visually:

• Look for a repeating segment: observe the graph between one peak to the next peak, or one trough to the next trough.
• In the graph of \(y = \tan(x)\), each segment or cycle from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) represents one complete period.
• Recognize that this repeating nature allows the tangent to return to its initial value after each period.

Understanding the period aids in predicting the function's behavior over any interval.

Asymptotes in Trigonometry

Asymptotes give us important insights into the behavior of the tangent function. Particularly, they show where the function exhibits discontinuities as it approaches infinity or negative infinity. For \(y = \tan(x)\), these vertical asymptotes occur at \(x = n\pi + \frac{\pi}{2}\) for any integer \(n\).
These lines are crucial because:

• They indicate points where the tangent function has undefined values due to division by zero in its calculation.
• Graphically, the lines appear where the curve seems to shoot upward or downward without bound.
• As \(x\) nears these values, \( \tan(x) \) increases or decreases dramatically, leading to the typical gap seen on its graph.

Recognizing the position and impact of these asymptotes is key in graphing and understanding tangent functions properly.