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Chapter 2: Problem 48

Sketch at least one cycle of the graph of each function. Determine the period and the equations of the vertical asymptotes. $$ y=\tan \left(\frac{\pi}{4} x+\frac{3 \pi}{4}\right) $$

Short Answer

Expert verified
The period is 4, and vertical asymptotes are at x = 4k - 2.

Step by step solution

01

- Identify the Tangent Function

Recognize that the function given is a tangent function, which is periodic and has vertical asymptotes where the function is undefined.
02

- Determine the Period of the Tangent Function

The period of the general tangent function \(y = \tan(bx + c)\) is \(\frac{\pi}{|b|}\). Here, \(b = \frac{\pi}{4}\), so the period is \(\frac{\pi}{\left| \frac{\pi}{4} \right|} = 4\).
03

- Find the Argument of the Tangent Function

The function becomes undefined when \(\tan\left( \frac{\pi}{4} x + \frac{3 \pi}{4} \right) = \pi/2 + k\pi, k \in \mathbb{Z} \). To find where these asymptotes occur, set the argument equal to the undefined points: \[\frac{\pi}{4} x + \frac{3 \pi}{4} = \frac{\pi}{2} + k \pi \]
04

- Solve for the Vertical Asymptotes

Solving the equation for \(x\): \[\frac{\pi}{4} x + \frac{3 \pi}{4} = \frac{\pi}{2} + k \pi\frac{\pi}{4} x = \frac{\pi}{2} + k \pi - \frac{3 \pi}{4}\frac{\pi}{4} x = \frac{2 \pi}{4} + \frac{4 k \pi}{4} - \frac{3 \pi}{4}\frac{\pi}{4} x = \frac{\pi + 4k\pi - 3\pi}{4}\frac{\pi}{4} x = \frac{(4k - 2)\pi}{4}\frac{\pi}{4} x = (k - \frac{1}{2}) \pi\ x = 4(k - \frac{1}{2})\ x = 4k - 2\] Therefore, the vertical asymptotes occur at \(x = 4k - 2, k \in \mathbb{Z}\).
05

- Sketch the Graph

To sketch the graph of the function \(y = \tan \left( \frac{\pi}{4} x + \frac{3 \pi}{4} \right) \), draw vertical asymptotes at \(x = -2, x = 2, x = 6, \) (one period). The graph crosses the x-axis halfway between each pair of asymptotes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trigonometric_graphs
Trigonometric functions such as sine, cosine, and tangent are periodic and form specific patterns on a coordinate plane. To graph these functions, we follow certain key rules:
  • Identify key points: Start by identifying where the function is undefined and where it crosses the x and y axes.

  • Period and Phase: Determine the period (the distance before the function repeats) and any phase shifts (horizontal displacements).

  • Vertical Asymptotes: Specifically for the tangent function, locate the vertical asymptotes (where the function becomes undefined). These will be the lines the function approaches but never touches.

  • Complete one cycle: Sketch at least one full cycle including where the function intersects the axes and where the vertical asymptotes occur.

  • Repetition: After graphing one cycle, replicate the pattern for additional cycles.

    • By understanding and following these steps, one can accurately graph any trigonometric function.

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Most popular questions from this chapter

Describe the graph of each function then graph the function between -2 and 2 using a graphing calculator or computer. $$ y=\sin \pi x+\sin \frac{\pi}{2} x $$

Determine the amplitude, phase shift, and range for each function. Sketch at least one cycle of the graph and label the five key points on one cycle as done in the examples. $$ y=-\cos x $$

Determine the amplitude, phase shift, and range for each function. Sketch at least one cycle of the graph and label the five key points on one cycle as done in the examples. $$ y=\frac{1}{2} \cos x $$

Let \(f(x)=\tan (x), g(x)=x+3,\) and \(h(x)=2 x .\) Find the following. $$ f(g(h(x))) $$

Solve each problem. If the frequency of a sine wave is 40,000 cycles per second, then what is the period?
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