Question

ERROR ANALYSIS Describe and correct the error in describing the transformation of \(f(x)=\tan x\) represented by \(g(x)=2 \tan 5 x\).

Short answer

The function \(g(x)=2 \tan 5x\) represents a vertical stretch of the \(\tan x\) function by a factor of 2 and a horizontal compression by a factor of 1/5. Without knowing what the original incorrect description was, it can only be concluded that the error involved a misunderstanding of these transformations.

Step-by-step solution

Identify Error in the Original Description

The first step in this problem is to identify the error that was made in the transformation of the function \(f(x) = \tan x\) to \(g(x) = 2 \tan 5x\). Without the original error mentioned, let's proceed with the correct transformation.

Describe the Vertical Stretch

You can first notice the coefficient '2' in the function \(g(x) = 2 \tan 5x\). The number 2 is a positive value that is bigger than 1. This means it vertically stretches the function by a factor of 2. In other words, the y-values of the original graph are multiplied by 2, making the graph appear 'steeper' or 'elongated' vertically.

Describe the Horizontal Compression

Next, take a look at the '5' within the tan function. This number signifies a horizontal compression of the function by a factor of 1/5. In other words, the x-values of the original graph are divided by 5, causing the period of the graph to decrease by a factor of 5.

Key concepts

Vertical Stretch

When we talk about a vertical stretch in the context of trigonometric functions, we are referring to a transformation that alters the amplitude of the function. This means that every point on the graph is pulled away from or pushed towards the x-axis, depending on the multiplication factor. For instance, the transformation of the function \(f(x) = \tan x\) to \(g(x) = 2 \tan x\) includes a vertical stretch.

A multiplication factor greater than 1, such as 2, stretches the graph vertically. The effect of a vertical stretch is to make the peaks and valleys of the graph farther from the x-axis, effectively increasing the amplitude. In the given function \(g(x) = 2 \tan 5x\), each of the y-values from the original tangent function \(f(x)\) is doubled, causing the graph to elongate upwards and downwards. This transformation doesn't affect the x-values and therefore does not alter the period or the phase shift of the tangent function.

Horizontal Compression

Horizontal compression is observed when a trigonometric function is transformed by multiplying the variable by a constant greater than one. The effect of this transformation is to 'squeeze' the graph horizontally, reducing its period. In the example \(g(x) = 2 \tan 5x\), the constant 5 compresses the standard tangent function horizontally by a factor of \(\frac{1}{5}\).

This means that each x-value of the original function is multiplied by 5, causing the function to complete one full cycle in one-fifth of the usual space. As a result, the peaks and valleys that formerly existed within a certain interval along the x-axis are now encountered more frequently, resulting in a graph that exhibits more cycles over any given x-interval. Importantly, the horizontal compression does not affect the y-values, and thus, the vertical stretch remains intact during this transformation.

Tangent Function Transformations

Transformations of the tangent function can include shifts, stretches, compressions, and reflections. Each of these transformations alters the original function \(f(x) = \tan x\) in a unique way. For the transformation to \(g(x) = 2 \tan 5x\), we are dealing with a vertical stretch and a horizontal compression, without any shifts or reflections.

The tangent function, known for its periodicity and asymptotes, is sensitive to these changes. A vertical stretch multiplies the y-values, affecting the 'steepness' of the graph without affecting the x-values. On the other hand, horizontal compression multiplies the x-values inside the tangent function, effectively reducing the period of the function and changing how quickly the function repeats its pattern. Understanding these transformations is crucial because it allows one to predict and draw the behaviour of the transformed function based on the behaviour of the original function.

Graphing Tangent Functions

Graphing tangent functions, especially when they've been transformed, requires an understanding of both their original properties and the effects of their transformations. The basic graph of \(f(x) = \tan x\) has a period of \(\pi\), with asymptotes occurring at \(\frac{1}{2}\pi + k\pi\), where k is an integer.

To graph a transformed tangent function like \(g(x) = 2 \tan 5x\), you start by graphing the basic tangent function. Then apply the transformations step by step. A vertical stretch by a factor of 2 will double the distance of all points from the x-axis. Next, a horizontal compression by a factor of 5 will reduce the period to \(\frac{\pi}{5}\), meaning that the function will complete a full cycle every \(\frac{\pi}{5}\) units along the x-axis.

Drawing this graph accurately requires careful plotting of the new asymptotes and ensuring the stretched amplitude is reflected in the graph's peaks and troughs. One should also be mindful that the graph stretches infinitely in both vertical directions, illustrating the tangent function's characteristic of having no maximum or minimum values.