/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 15

Sketch the graph of the function. (Include two full periods.) $$y=\frac{1}{3} \tan x$$

Short Answer

Expert verified
The graph of the function \(y = \frac{1}{3} \tan x\) is a wave-like shape, repeating every \(\pi\) along the x axis, with asymptotes at every odd multiple of \(\pi/2\). The graph is less steep than the basic tangent function because of the scalar \(\frac{1}{3}\).

Step by step solution

01

Understand the Tan Function

The first step is to understand what the basic \(\tan\) function shape is like. Tangent function has a period of \(\pi\),, it starts at \(0\) and stretches infinitely at \(\pi/2\) and \(-\pi/2\) (these points are called asymptotes). The graph repeats after every \(\pi\) in both directions of the x axis. This is a very important fact when sketching the tangent graph.
02

Apply the Scalar

The next step is to understand the scalar \(\frac{1}{3}\). Here, it affects the vertical stretch of the graph by reducing the rate of increase or decrease of the y-values. Therefore, the graph will be less steep as compared to the graph of basic tangent function, \(\tan x\).
03

Sketch the Graph

Now we understand the basic shape of a tangent function and the impact of the scalar \(\frac{1}{3}\), begin sketching the function \(y = \frac{1}{3} \tan x\). Because two full periods are required, draw the graph from \(-2\pi\) to \(2\pi\). Remember to include the vertical asymptotes at every odd multiple of \(\pi/2\), and show that graph is less steep than the usual \(\tan x\) graph because of the scalar \(\frac{1}{3}\).
04

Verify

Take a few values of x and calculate the corresponding y-values. Plot these points on the graph to ensure that the drawn graph is accurate.

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