Problem 36 - Precalculus Mathematics for Calculus

Question

Find the period and graph the function. $$y=2 \tan \frac{\pi}{2} x$$

Step-by-Step Solution

Answer

The period is 2, and the function repeats every 2 units on the x-axis.

Step 1: Identify the Standard Form of Tangent Function

The standard form of the tangent function is \( y = a \tan(bx) \). In our function, \( y = 2 \tan \left( \frac{\pi}{2} x \right) \), we can identify \( a = 2 \) and \( b = \frac{\pi}{2} \).

Step 2: Calculate the Period

The period of the tangent function \( y = a \tan(bx) \) is given by the formula \( \text{Period} = \frac{\pi}{|b|} \). In this case, \( b = \frac{\pi}{2} \), so the period is \( \text{Period} = \frac{\pi}{\left| \frac{\pi}{2} \right|} = \frac{\pi}{\pi} \cdot 2 = 2 \). Thus, the period of the function is 2.

Step 3: Graph the Function

To graph \( y = 2 \tan \left( \frac{\pi}{2} x \right) \), first note that this graph will repeat every 2 units along the x-axis, based on its period. The tangent function has vertical asymptotes where \( \frac{\pi}{2} x = \frac{\pi}{2} + k\pi \), simplifying to \( x = 1 + 2k \) for integer \( k \). Plot these asymptotes and then sketch the tangent curve which passes through the origin (since \( y = 0 \) when \( x = 0 \)) and repeats every period.

Key Concepts

Period of a Function

The period of a function is how long it takes for the function to repeat its pattern. When dealing with trigonometric functions like sine, cosine, and tangent, understanding the period is essential.

For a standard tangent function of the form \( y = a \tan(bx) \), the period can be calculated using the formula:

• \( \text{Period} = \frac{\pi}{|b|} \)

In our specific function, \( y = 2 \tan \left( \frac{\pi}{2} x \right) \), we have \( b = \frac{\pi}{2} \).
Calculating the period gives:

• \( \text{Period} = \frac{\pi}{ \left| \frac{\pi}{2} \right| } = \frac{\pi}{\pi} \cdot 2 = 2 \)

This means that the function repeats itself every 2 units along the x-axis.

By knowing the period, you can easily sketch and understand the general layout of the function on a graph.

Vertical Asymptotes

When graphing tangent functions, vertical asymptotes play a crucial role. These are lines the graph approaches but never touches or crosses. They're like invisible walls guiding the curve.

For the function \( y = 2 \tan \left( \frac{\pi}{2} x \right) \), vertical asymptotes occur where the function evaluation "blows up" or becomes undefined. In simpler terms, we're interested in the values of x when the denominator of the trigonometric argument becomes zero:

The condition for vertical asymptotes in tangent functions is:

• \( \frac{\pi}{2} x = \frac{\pi}{2} + k\pi \)

This equation simplifies to:

• \( x = 1 + 2k \)

where \( k \) is any integer (0, ±1, ±2, ...). These x-values are where our vertical asymptotes lie.

Plotting these on the graph helps to visually separate each repeated section of the curve.

Tangent Function

The tangent function is one of the basic trigonometric functions and is particularly interesting due to its distinctive properties. Unlike sine and cosine, which are periodic and bounded, the tangent function is periodic and unbounded, meaning it keeps increasing and decreasing between each repeat.

The general form for a tangent function is \( y = a \tan(bx) \). The coefficient \( a \) affects the stretch or compression in the y-direction, while \( b \) affects the period of the function as seen earlier. In our case, \( y = 2 \tan \left( \frac{\pi}{2} x \right) \) indicates a vertical stretch by a factor of 2.

Tangent functions have several distinctive characteristics:

• Their graphs consist of repeating curves with vertical asymptotes and points of inflection.
• The fundamental cycle of the tangent curve starts at the point (0,0) and repeats every period — in this case, every 2 units along the x-axis.
• Tangent graphs do not have maximum or minimum values, reflecting their unbounded nature.

Understanding and visualizing these properties will help in mastering their graphing and behavior on real-life applications.