Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed. $$ y=\tan x $$

Step-by-Step Solution

Answer

The period is \(\pi\), no amplitude, and no shifts.

Step 1: Identify the Function Type

The function given is a tangent function, denoted as \( y = \tan x \). The tangent function is periodic and has certain characteristics that we need to determine like period, amplitude, and shifts.

Step 2: Determine the Period

For the tangent function \( y = \tan x \), the period is typically \( \pi \) because it repeats every \( \pi \) units.

Step 3: Amplitude Consideration

Unlike sine and cosine functions which have amplitudes, the tangent function doesn't have a defined amplitude because it increases and decreases without bound.

Step 4: Shifts Identification

In the equation \( y = \tan x \), there are no horizontal or vertical shifts. The function is centered at the origin and there are no added constants that suggest any transformation.

Step 5: Graphing the Function

To graph \( y = \tan x \) over the interval \(-5 \leq x \leq 5\), observe that the function has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \). The function will pass through zeros at integer multiples of \( \pi \). The branches of tangent between these asymptotes increase from \(-\infty\) to \(+\infty\).

Key Concepts

Periodicity

In mathematics, periodicity refers to the property of a function to repeat its values at regular intervals, known as periods. For the tangent function, which is given as \( y = \tan x \), periodicity is a fundamental characteristic. The tangent function is periodic with a period of \( \pi \). This means that the function repeats its pattern every \( \pi \) units along the x-axis.

With that in mind, if you start at any point \( x \) and move \( \pi \) units forward (or backward), the value of \( \tan x \) will be the same. For instance, \( \tan( x ) = \tan( x + \pi ) = \tan( x - \pi ) \). This periodicity results in a repeating pattern along the graph, making the study of trigonometric functions both predictable and fascinating.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visually represent the relationships defined by the function. For \( y = \tan x \), graphing this function provides insight into its behavior.

When graphing \( y = \tan x \) over the interval \(-5 \leq x \leq 5\), it's crucial to identify key features:

• Vertical asymptotes are present at odd multiples of \( \frac{\pi}{2} \). These are points where the function approaches infinity and thus, there are no outputs.
• The graph crosses the x-axis at integer multiples of \( \pi \), where the function equals zero.

Between the asymptotes, the graph of \( \tan x \) rises sharply from \(-\infty\) to \( +\infty \). This behavior offers a clear and repetitive wave-like pattern, thanks to the periodicity of the function.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. Besides tangent, the main trig functions include sine and cosine. Tangent, or \( \tan(x) \), can technically be defined as the ratio of the sine to the cosine of an angle:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

This relation means that wherever \( \cos(x) = 0 \), \( \tan(x) \) becomes undefined due to division by zero, hence the vertical asymptotes in its graph.

Understanding these trigonometric functions helps one grasp complex mathematical cycling behaviors. They are foundational in fields of study such as physics, engineering, and computer science, where oscillations and wave patterns are prevalent.

Function Transformations

Function transformations involve shifting or stretching the graph of a function, changing its appearance without altering the essence of the function. Common transformations include translations, reflections, and dilations.

For \( y = \tan x \), the basic form does not inherently involve any transformations:

• Horizontal shifts would be indicated by adding or subtracting a constant inside the function argument.
• Vertical shifts are indicated by adding or subtracting a constant directly to the function.

In the problem \( y = \tan x \), no constants are added or subtracted, so there are no horizontal or vertical shifts. Similarly, there's no stretching or shrinking occurred since there are no multiplicative factors involved. This lack of transformation keeps the function in its purest trigonometric form, centered at the origin without distortion.