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Magazine Chapter 5: Problem 10
Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\tan \left(x+\frac{\pi}{2}\right)$$
Short Answer
Expert verified
The period is \( \pi \), and the vertical asymptotes are at \( x = n\pi \).
Step by step solution
01
Identify the function
The function given is a trigonometric function, specifically the tangent function: \( y = \tan\left(x + \frac{\pi}{2}\right) \). This is a transformed version of the basic \( y = \tan(x) \) function.
02
Standard period of tangent
The standard period of the tangent function \( y = \tan(x) \) is \( \pi \). This means that the tangent function repeats every \( \pi \) units along the x-axis.
03
Determine the shifted period
The transformation \( x + \frac{\pi}{2} \) indicates a horizontal shift, but it does not affect the period of the function. Therefore, the period of \( y = \tan(x + \frac{\pi}{2}) \) is still \( \pi \).
04
Identify vertical asymptotes
The original tangent function has vertical asymptotes where \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. For \( y = \tan\left(x + \frac{\pi}{2}\right) \), this shifts the asymptotes to \( x = n\pi \).
05
Sketch the graph
To sketch the graph of \( y = \tan\left(x + \frac{\pi}{2}\right) \), draw vertical lines at \( x = n\pi \) to represent the asymptotes. Between each pair of asymptotes, the graph will have an S-shape, similar to the standard tangent function but shifted left by \( \frac{\pi}{2} \). The curve will pass through zero at points like \( x = -\frac{\pi}{2}, \frac{\pi}{2}, ... \), and will ascend from negative to positive infinity across each period.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Period of Tangent Function
Every trigonometric function has a specific period that indicates how often the function repeats itself along the x-axis. For the tangent function, this period is especially important to grasp since it helps determine where the function starts and finishes its repetition.
- The basic tangent function, written as \( y = \tan(x) \), has a period of \( \pi \).
- This means that the graph of the tangent function will repeat every \( \pi \) units.
- For any transformed tangent function of the form \( y = \tan(bx + c) \), the period is computed by evaluating \( \frac{\pi}{|b|} \), where \( b \) is the coefficient of \( x \).
Graphing Trigonometric Functions
Graphing trigonometric functions is fundamental in understanding their behavior and properties. Let's simplify how we can graph these functions.
- Start by understanding the basic shape. For the tangent function \( y = \tan(x) \), it has an S-shape or can be described as increasing between asymptotes.
- Identify its period 鈥� for the tangent, it鈥檚 \( \pi \). This dictates the distance between repeating sections of the graph.
- Recognize the function undergoes transformations like shifts or stretches/compressions, which modify its graph accordingly.
Vertical Asymptotes
Vertical asymptotes of a function occur where the function value becomes undefined. For the tangent function, vertical asymptotes play a crucial role in shaping the graph鈥檚 behavior.
- The formula for asymptotes in the basic tangent function, \( y = \tan(x) \), is \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.
- These lines show up as vertical lines where the tangent function's graph shoots off towards positive or negative infinity.
Function Transformation
Function transformation is all about shifting, stretching, or compressing a function's graph in relation to its axes. It enables a standard function to take different forms and positions.
- A horizontal shift results from adjusting the \( x \) part of the function. For instance, \( y = \tan(x + \frac{\pi}{2}) \) signifies a shift to the left by \( \frac{\pi}{2} \).
- Vertical stretches/compressions are managed by multiplying the entire function by a constant. Although not present in our example, this alters how steep the graph appears.
