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Chapter 5: Problem 37

Find the period and graph the function. $$y=\tan 2\left(x+\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The period is \( \frac{\pi}{2} \) and the graph shifts \( \frac{\pi}{2} \) units left.

Step by step solution

01

Identify the Standard Form

The standard form of a tangent function is given by \( y = \tan(bx+c) \), where the period is calculated as \( \frac{\pi}{|b|} \). In the given function, \( y = \tan(2(x+\frac{\pi}{2})) \). Rewrite it as \( y = \tan(2x + \pi) \), identifying \( b = 2 \) and \( c = \pi \).
02

Calculate the Period of the Function

Using the formula for the period of a tangent function: \( \frac{\pi}{|b|} \), substitute \( b = 2 \) into the formula. Therefore, the period is \( \frac{\pi}{2} \). This means that the function will repeat every \( \frac{\pi}{2} \) units along the x-axis.
03

Determine the Phase Shift

The phase shift of the function can be found using the formula \( -\frac{c}{b} \). Substitute \( c = \pi \) and \( b = 2 \) into the formula, giving \( -\frac{\pi}{2} \). This indicates the graph is shifted \( \frac{\pi}{2} \) units to the left.
04

Sketch the Graph of the Function

Draw the graph of \( y = \tan 2x \) first, which has a period of \( \frac{\pi}{2} \). Then apply the phase shift of \( \frac{\pi}{2} \) units to the left. This involves shifting the entire graph (including its asymptotes and repeating cycles) \( \frac{\pi}{2} \) units in the negative direction on the x-axis.

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Key Concepts

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

Period of a Function
The period of a function tells us how often the function repeats itself along the x-axis. For the tangent function, which is different from sine and cosine functions, the period is given by the formula:
  • Period = \( \frac{\pi}{|b|} \)
Here, \( b \) is the coefficient of \( x \) within the function's argument. Unlike the \( 2\pi \) period seen in sine and cosine, tangent functions naturally repeat every \( \pi \) (without any modification to \( b \)).
In this problem, since we have \( y = \tan 2(x+\frac{\pi}{2}) \), we first simplify it to \( y = \tan(2x + \pi) \).
  • We see that \( b = 2 \).
  • Then using our period formula, we find that the period for this function is \( \frac{\pi}{2} \).
This means every \( \frac{\pi}{2} \) along the x-axis, the function's graph will repeat itself, showing similar patterns of behavior.
Phase Shift
The phase shift of a trigonometric function is a horizontal movement of the entire graph along the x-axis. We can calculate this shift with the formula:
  • Phase Shift = \( -\frac{c}{b} \)
Where \( c \) is the constant added to or subtracted from \( bx \) within the function, and \( b \) is the multiplier of \( x \). In this exercise, the function is written as \( y = \tan(2x + \pi) \). Let's break it down:
  • Here, \( c = \pi \)
  • and \( b = 2 \).
  • Using the formula, the phase shift becomes \( -\frac{\pi}{2} \).
This shift indicates that the entire graph of the function \( y = \tan 2(x+\frac{\pi}{2}) \) will be moved \( \frac{\pi}{2} \) units to the left, compared to the graph of \( y = \tan 2x \).
This phase shift is essential because it aligns the function's cycle to start at a different point on the x-axis.
Tangent Function
The tangent function, one of the primary trigonometric functions, behaves distinctively compared to sine and cosine. Its general form is given by:
  • \( y = \tan(bx + c) \)
It's important to recognize its characteristics:
  • The tangent graph has asymptotes, which are vertical lines where the function is undefined and the graph approaches infinity.
  • Typically, without alterations, it has a period of \( \pi \), where it repeats its behavior between the intervals.
In this specific exercise, the function has been modified to \( y = \tan 2(x+\frac{\pi}{2}) \), affecting both its period and shift.
  • The period changes to \( \frac{\pi}{2} \), due to the coefficient \( b = 2 \).
  • The phase shift of \( -\frac{\pi}{2} \) results from \( c = \pi \).
  • These alterations mean the typical repeating cycle and asymptotes of the tangent function are adjusted to reflect \( \frac{\pi}{2} \) spacing along the x-axis.
Graphing it involves understanding these foundational changes, leading to predictions about where the function diverges and repeats.

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Most popular questions from this chapter

Graph the three functions on a common screen. How are the graphs related? $$y=\cos 3 \pi x, \quad y=-\cos 3 \pi x, \quad y=\cos 3 \pi x \cos 21 \pi x$$

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$\left(\frac{3}{5}, \frac{4}{5}\right)$$

Find the value of each of the six trigonometric functions (if it is defined) at the given real number \(t\). Use your answers to complete the table. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & \sin t & \cos t & \tan t & \csc t & \sec t & \cot t \\ \hline 0 & 0 & 1 & & \text { undefined } & & \\ \hline \frac{\pi}{2} & & & & & & \\ \hline \pi & & & 0 & & & \text { undefined } \\ \hline \frac{3 \pi}{2} & & & & & & \\ \hline \end{array}$$ $$t=\pi$$

Find the values of the trigonometric functions of \(t\) information. \(\tan t=\frac{1}{4}, \quad\) terminal point of \(t\) is in quadrant III

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=\frac{13 \pi}{6}$$
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