Problem 1 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of vertical translations to graph at least two cycles of the given functions. $$f(x)=\tan x-3$$

Step-by-Step Solution

Answer

Graph the function \( f(x) = tan(x) - 3 \) by shifting the standard tangent function three units down. Keep the same period, asymptotes, and x-intercepts, just shift all y-values three units down.

Step 1: Understand the base function

Recognize that the base function is the standard tangent function \(tan(x)\). This function has a period of \(π\), vertical asymptotes at \(x = ±π/2 + kπ\) for any integer \(k\), and it crosses the x-axis at \(x = kπ\) for any integer \(k\).

Step 2: Apply the vertical translation

A vertical translation of -3 means that every point on the base function will move 3 units downwards. So, the function \(f(x) = tan(x) - 3\) will have the same shape as \(tan(x)\), but all y-values will be reduced by 3, meaning the function is shifted downwards by 3 units.

Step 3: Plot the graph

Plot the function \(f(x) = tan(x) - 3\). First draw a dashed vertical line at each of the asymptotes to guide your sketch. Then, sketch two cycles of the graph by starting at the x-intercepts, heading towards the asymptotes, and moving 3 units below the x-axis because of the translation.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is a fundamental trigonometric function that arises from the sine and cosine functions. It can be expressed as the ratio between sine and cosine, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

The primary characteristics of the tangent function include:

• **Periodicity**: The tangent function is periodic with a standard period of \( \pi \). This means the function repeats its pattern every \( \pi \) units along the x-axis.
• **Asymptotes**: It possesses vertical asymptotes, occurring at every odd multiple of \( \frac{\pi}{2} \), i.e., \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. At these points, the function is undefined and appears to shoot off to infinity.
• **Intercepts**: The tangent function crosses the x-axis at multiples of \( \pi \), such as \( x = k\pi \) for integer \( k \).

The tangent curve steadily increases from \(-\infty\) to \(\infty\) within each period, beginning at \(-\frac{\pi}{2} \) and ending at \( \frac{\pi}{2} \) before repeating its cycle.

Periodic Functions

Periodic functions are those that repeat their values at regular intervals, or periods. The idea of periodicity is pivotal in various fields, especially in trigonometry and signal processing. These functions are essential to describe phenomena that cycle or repeat over time.

Some key features of periodic functions include:

• **Period**: This is the length of one complete cycle of the function. For example, \( \tan(x) \) has a period of \( \pi \).
• **Amplitude**: Typically applicable to sine and cosine, it refers to the height from the central axis to the peak of the wave. Tangent, however, does not have a defined amplitude because it does not have maximum or minimum values.
• **Phase Shift**: A horizontal translation of the graph that shifts the cycle along the x-axis without altering its shape.

The concept of periodicity allows us to predict and understand repeating patterns, making such functions valuable for modeling cycles found in natural and human-made systems.

Graphing Transformations

Graphing transformations involve altering the position and shape of the graph of a function. These changes can include translations, reflections, scalings, and rotations on the graph.

In the context of the tangent function or any trigonometric function, vertical translations are common transformations. They can be understood as moving the entire graph up or down the y-axis:

• **Vertical Translation**: A transformation where each point on the graph is moved up or down by a specific number of units. For example, in the function \( f(x) = \tan(x) - 3 \), every point of \( \tan(x) \) is shifted downward by 3 units.
• **Horizontal Translation**: Moving the graph left or right, generally not affecting the vertical asymptotes of the tangent function.
• **Reflection and Scaling**: Although not needed in the context of this specific exercise, these transformations can flip the graph or stretch and compress it along the axis.

By understanding and applying these transformations, one can predict how a function will visually appear on a graph, aiding in sketching and analyzing its behavior. This skill is particularly useful for students learning to manipulate trigonometric functions and beyond.