Problem 40 - Precalculus : Building Concepts and Connections

Question

Graph the given pair of functions in the same window. Graph at least two cycles of each function, and describe the similarities and differences between the graphs. $$f(x)=\tan (3 x) ; f(x)=3 \tan (x)$$

Step-by-Step Solution

Answer

The functions \(f(x) = \tan(3x)\) and \(f(x) = 3\tan(x)\) are both variations of the tangent function, but differ in terms of amplitude and period. The former has a smaller period, which makes it complete its cycle rapidly while the latter has a larger amplitude, resulting in its values rising more steeply. Both graphs share the similarity in shape being based off the tangent function.

Step 1: Graphing the first function

To graph the function \(f(x) = \tan(3x)\), one must first understand its characteristics. The period of \(f(x) = \tan(3x)\) is \(\pi/3\) as the coefficient of x in the tangent function is 3. Using this period, one marks points along the x-axis at intervals of \(\pi/6\), \(\pi/3\), \(2\pi/3\), and so on, and then sketches the graph over at least two cycles mentioned intervals.

Step 2: Graphing the second function

To graph the second function \(f(x) = 3\tan(x)\), one must note that the amplitude (or vertical stretch) is 3 and the period is \(\pi\), as there is no coefficient on the x inside the tangent function. Similar to step 1, one marks points along the x-axis at intervals of \(\pi/4\), \(\pi/2\), \(3\pi/4\), and so on, and sketches the graph over at least two cycles intervals.

Step 3: Comparing the two functions

Upon examining the graphs of the two functions, the main similarities and differences can be noted. Both functions have a similar shape since they are both based on the tangent function, but the first function \(f(x) = \tan(3x)\) completes its cycle more rapidly due to having a smaller period while the second function \(f(x) = 3\tan(x)\) has a larger amplitude due to the factor of 3 multiplying the tangent function.

Key Concepts

Graphing

Graphing trigonometric functions can initially seem complex, but by breaking down the steps, this process becomes quite intuitive. Let's look at how we can effectively graph the tangent function. The key to graphing a tangent function like \( f(x) = \tan(3x) \) is to understand its periodic nature. To start:

• Identify the period of the function. For \( \tan(3x) \), the usual period of \( \tan(x) \), which is \( \pi \), is divided by 3, making the period \( \frac{\pi}{3} \).
• Mark the asymptote boundaries, as tangent functions have vertical asymptotes where the function is undefined. For this function, these occur at multiples of \( \frac{\pi}{6} \), \( \frac{\pi}{3} \), \( \frac{2\pi}{3} \), etc.
• Plot key points between the asymptotes. The tangent function will pass through the origin (0,0) every sub-interval (middle of each period).
• Draw a curve representing the rapid increase or decrease of \( \tan(x) \) near the asymptotes.

Remember to complete at least two cycles to capture the periodic nature fully. Each cycle of the tangent function repeats the distinct S-shape we see across its domain.

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or periods. Understanding their behavior is crucial when studying trigonometric functions like sine, cosine, and tangent. With the tangent function, which we are examining:

• The standard period for \( \tan(x) \) is \( \pi \). This is the interval over which the pattern repeats.
• For functions like \( f(x) = \tan(3x) \), the coefficient '3' modifies the period to \( \frac{\pi}{3} \), meaning the function's shape repeats more frequently over smaller intervals.
• Periodic functions often exhibit both vertical and horizontal translations, depending on additional constants, but basic transformations focus on period changes.
• Vertical stretching or compression, like in \( 3\tan(x) \), affects the y-values but does not alter the period.

Understanding these properties aids in recognizing patterns across the function's graph, making predicting behavior easier.

Tangent Function

The tangent function, \( \tan(x) \), is one of the primary trigonometric functions. It is unique because it can model phenomena with vertical asymptotes, reflecting its undefined nature at certain points. Key aspects of the tangent function include:

• The tangent function can increase without bound, approaching infinity as it nears its vertical asymptotes, which occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
• Unlike sine and cosine, the tangent function does not have a maximum or minimum value; instead, its values range from negative to positive infinity.
• Both \( \tan(3x) \) and \( 3\tan(x) \) retain this behavior but with variations in period and amplitude respectively. \( \tan(3x) \) has a shorter cycle, while \( 3\tan(x) \) is stretched vertically.
• This function is periodic and odd, meaning it is symmetric about the origin and repeats every period.

These characteristics make \( \tan(x) \) a powerful function for modeling periodic phenomena with steep, repeating changes, such as in wave forms or oscillations.