Problem 1 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=4 \tan x$$

Step-by-Step Solution

Answer

The period of the function is \( \pi \). Asymptotes are located at \( x = \frac{\pi}{2} + k\pi \).

Step 1: Identify the function type and form

The given function is of the form \( y = a \tan(bx + c) + d \). In this function, \( a = 4 \), \( b = 1 \), \( c = 0 \), and \( d = 0 \). This tells us it is a tangent function where \( a \), the amplitude, affects the steepness, but not the period.

Step 2: Determine the period

The period of a basic tangent function \( \tan(bx) \) is influenced by \( b \). The formula for the period of \( \tan(bx) \) is \( \frac{\pi}{b} \). In this case, \( b = 1 \), so the period is \( \frac{\pi}{1} = \pi \).

Step 3: Chart the general behavior and identify asymptotes

Tangent functions have vertical asymptotes where they are undefined, which occur at \( x = \frac{\pi}{2} + k\pi \) for integers \( k \). For \( y = 4 \tan(x) \), vertical asymptotes are at \( x = \frac{\pi}{2} + k\pi \).

Step 4: Sketch the graph with the determined characteristics

Start by plotting the vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = -\frac{\pi}{2} \). Between these asymptotes, draw the tangent curve which crosses \( y=0 \) at \( x=0 \). Increase the steepness due to \( a = 4 \), meaning the curve will appear steeper compared to \( \tan x \). Repeat the pattern for a few periods to show several cycles.

Key Concepts

Tangent Function

The tangent function, typically expressed as \( y = \tan(x) \), is one of the basic trigonometric functions. Unlike sine and cosine, which have a wave-like appearance, the tangent function has a different kind of behavior. Its graph is highly distinctive with recurring vertical asymptotes. These asymptotes lead to the function being undefined at certain points.
What makes the tangent function interesting is its odd symmetry. It is an odd function, meaning it has rotational symmetry about the origin. The tangent starts from zero, moves upwards towards infinity, and then drops back down from negative infinity to zero again between each set of asymptotes.

• Tangent functions repeat every \( \frac{\pi}{b} \), where \( b \) determines the frequency.
• They pass through the origin, so \( y = \tan(x) \) equals zero at \( x = 0 \), \( x = \pi \), \( x = 2\pi \), and so on.
• For the function \( y = 4\tan(x) \), the steepness or slope of the curve is increased, making it look much steeper compared to a regular \( \tan(x) \).

Period of Trigonometric Functions

The period of a trigonometric function is the length of the interval over which the function repeats itself. For the tangent function, the period is particularly straightforward.
The basic formula to find the period of a tangent function is \( \frac{\pi}{b} \), where \( b \) is the coefficient of \( x \) in \( bx \). In the standard \( y = \tan(x) \), \( b = 1 \), so the period is \( \pi \). This means the function repeats every \( \pi \) units along the x-axis.

• For example, for the function \( y = 4 \tan(x) \), because \( b = 1 \), the period remains \( \pi \).
• The period tells where the graph of the function repeats its patterns and oscillations.

Understanding the period is fundamental when graphing these functions, as it dictates the "spacing" of the cycles. Recognizing the repetition helps in plotting multiple cycles accurately.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. They play a crucial role in defining the behavior of the tangent trigonometric function. For \( y = \tan(x) \), vertical asymptotes occur at points where the function is undefined. These are specifically located at \( x = \frac{\pi}{2} + k\pi \), with \( k \) being any integer.
These vertical asymptotes are crucial in sketching the graph because they act as the boundaries where the tangent function shifts from positive infinity to negative infinity. As the tangent graph approaches these lines, it turns incredibly steep and never crosses them.

• To graph \( y = 4 \tan(x) \), mark the vertical asymptotes at points like \( x = \frac{\pi}{2} \), where \( y \) is undefined.
• Between the asymptotes, the graph of the tangent will make a steep s-curve, passing through the origin.
• Vertical asymptotes occur every \( \pi \) units along the x-axis for this function.

Amplitude and Period

In trigonometric functions, amplitude traditionally refers to the maximum displacement from the center line of a wave-like graph, typically seen in sine and cosine functions. However, in the tangent function context, the term amplitude commonly reflects the "steepness" or the vertical stretch of the graph, influenced by the factor \( a \) in the equation \( y = a \tan(bx + c) + d \).
For \( y = 4 \tan(x) \), the '4' indicates that each cycle of the tangent will be four times as steep compared to \( y = \tan(x) \). Despite the increase in steepness, the period of the function remains unchanged as both are determined separately by \( b \).

• The period of \( y = a \tan(bx) \) depends purely on \( b \), calculated as \( \frac{\pi}{b} \).
• The amplitude or steepness changes only how steep the slope of the tangent curve is between two consecutive asymptotes.
• The function will repeat its pattern every \( \pi \) units, making it periodic regardless of its amplitude.

Understanding these relationships is key to mastering the sketching and analysis of tangent trigonometric functions.