Problem 17 - Algebra 2

Question

Sketch the graph of each tangent curve in the interval from 0 to 2\(\pi\) $$ y=\tan \frac{2 \pi}{3} \theta $$

Step-by-Step Solution

Answer

The graph of the function \( y = \tan \frac{2 \pi}{3} \theta \) will have asymptotes at \(\theta = \frac{3 k \pi}{4}\), where k is an integer. The pattern of the tangent curve, that resembles an 's', would repeat in each section between the asymptotes from 0 to 2\(\pi\).

Step 1: Understanding the Tangent Function

The tangent function repeats every \(\pi\) (180 degrees) and has vertical asymptotes where the function is undefined, which are at odd multiples of \(\frac{\pi}{2}\).

Step 2: Apply This Understanding to the Given Function

In the given function \( y = \tan \frac{2 \pi}{3} \theta \), the period of the function is changed. The coefficient of \(\theta\) inside the tangent function changes its period from \(\pi\) to \(\frac{3}{2 \pi}\). This means the function will repeat at every \(\frac{3}{2 \pi}\). So it will have asymptotes when \(\frac{2 \pi}{3}\theta\) is an odd multiple of \(\frac{\pi}{2}\). That means \(\theta = \frac{3 k \pi}{4}\), where k is an integer, will be asymptotes.

Step 3: Sketch the Graph

Using the obtained information from Steps 1 and 2, the graph of the function can be sketched. Firstly, plot the asymptotes at \(\theta = \frac{3 k \pi}{4}\), where k is an integer. Then, sketch the typical shape of the tangent curve, the shape that looks like 's', in each section between the asymptotes from 0 to 2\(\pi\).

Key Concepts

Period of Trigonometric Functions

The period of a trigonometric function tells us how often the function repeats its values. For the familiar tangent function, an interesting feature is its unpredictable rise and fall. Unlike sine and cosine functions, whose periods are dictated by their complete wave cycles, the tangent has a basic period of \(\pi\) (180 degrees).
This difference is because the tangent repeats after every two quadrants, which equals \(\pi\) radians. Now, when we have a tangent function with a coefficient in front of \(\theta\), such as in \(y = \tan\left(\frac{2\pi}{3}\theta\right)\), this coefficient alters the period length.

• To find the new period, divide the standard period \(\pi\) by the coefficient of \(\theta\).
• Thus, for \(y = \tan\left(\frac{2\pi}{3}\theta\right)\), the period is \(\frac{3\pi}{2}\).

This means the function repeats every \(\frac{3\pi}{2}\), which changes how frequently the tangent curve completes its characteristic cycle.

Asymptotes

An asymptote is a line that a graph gets closer to, but never actually touches or crosses. In the context of the tangent function, asymptotes are points where the function is undefined and the graph heads towards infinity. These occur at each odd multiple of \(\frac{\pi}{2}\).
For the altered tangent function \(y = \tan\left(\frac{2\pi}{3}\theta\right)\), the presence of the \(\frac{2\pi}{3}\) coefficient changes where these asymptotes appear.

• To find them, set \(\frac{2\pi}{3}\theta\) equal to the odd multiples of \(\frac{\pi}{2}\).
• Solving this, we find the asymptotes occur at \(\theta = \frac{3k\pi}{4}\), where \(k\) is any integer.

When sketching the tangent graph, it's important to note these vertical asymptotes, as these are the locations where the graph will demonstratively approach but will not cross or reach.

Trigonometric Graph Sketching

Sketching the graph of a trigonometric function like \(y = \tan\left(\frac{2\pi}{3}\theta\right)\) is a step-by-step process. The goal is to represent all characteristics of the function within a specified interval, which for this exercise is from 0 to \(2\pi\).

• First, identify the period and much like we determined, it's \(\frac{3\pi}{2}\).
• Then, plot the vertical asymptotes at \(\theta = \frac{3k\pi}{4}\). These asymptotes divide the graph into intervals where the tangent function is defined.
• Within each interval, draw the characteristic arc shape of the tangent function, which typically resembles an elongated 's'.

These arcs should delicately approach each asymptote without crossing it. With these simple steps, you've sketched the graph, visualizing how it behaves differently from standard sine or cosine curves, maintaining clarity in its distinctive, repeating pattern.