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Magazine Chapter 9: Problem 71
Graph each function over a two-period interval. $$y=-1+2 \tan x$$
Short Answer
Expert verified
Graph \( y = -1 + 2 \tan x \) by sketching between \( -\pi \) and \( \pi \), taking note of vertical asymptotes and the downward shift.
Step by step solution
01
Determine the period of the function
The function given is a tangent function, \( y = -1 + 2 \tan x \). The standard period of the tangent function \( \tan x \) is \( \pi \). Thus, the period of \( 2 \tan x \) also remains \( \pi \). We need to graph over a two-period interval, which is \( 2\pi \).
02
Identify key features of the function
The amplitude of a tangent function does not affect its period, but the vertical stretch is doubled to \( 2 \) in the function \( 2 \tan x \). The function is shifted vertically downward by 1 unit because of the \(-1\) (i.e., the graph of \( -1+2\tan x \) is obtained by shifting \(2\tan x \) down by 1 unit).
03
Choose the domain for two periods
To cover two periods, work with the interval from \(-\pi\) to \(\pi\) (one full period from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) and another full period from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\)). This interval will allow us to chart the main features of the function twice.
04
Evaluate points on the graph
Evaluate key points such as \( x = 0 \), \( x = \pm \frac{\pi}{4} \), \( x = \pm \frac{\pi}{2} \), etc., within two periods. Observing that \( \tan(0) = 0 \), \( \, \tan\left(\frac{\pi}{4}\right)=1 \), and \( \tan\left(-\frac{\pi}{4}\right)=-1 \), calculate: \( y\) at \( x = 0 \) is \( -1+2\times0 = -1 \), at \( x = \frac{\pi}{4} \) is \( -1 + 2 \times 1 = 1 \), and at \( x = -\frac{\pi}{4} \) is \( -1 + 2\times(-1) = -3 \). The function has vertical asymptotes at \( x = \pm\frac{\pi}{2} \).
05
Sketch the graph
Plot the points calculated in the previous steps and sketch the curve, noting the vertical asymptotes at \( x = \pm \frac{\pi}{2} \) and \( x = \frac{3\pi}{2}, -\frac{3\pi}{2} \). The graph should show the periodicity, the intercepts, and the shifts, completing the cycle twice over from \( -\pi \) to \( \pi \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Periodicity
Periodicity is a key feature of trigonometric functions like the tangent function. Periodicity means that a function repeats its values at regular intervals, called periods. For the tangent function, the standard period is \( \pi \). This means that the graph of \( \tan x \) repeats every \( \pi \) units along the x-axis. In the case of our function \( y = -1 + 2 \tan x \):
- The period remains \( \pi \), despite the vertical stretch and shift.
- When graphing the function over a two-period interval, we lay out the graph from \(-\pi\) to \( \pi\).
Vertical Asymptotes
Vertical asymptotes occur where a function's value heads towards infinity. They are crucial in graphing because they represent boundaries where the function doesn鈥檛 exist. For the tangent function \( \tan x \), vertical asymptotes happen at every odd multiple of \( \frac{\pi}{2} \). This is due to the nature of the tangent function, which becomes undefined at these points. In our function \( y = -1 + 2 \tan x \):
- Vertical asymptotes appear at \( x = \pm\frac{\pi}{2} \), \( x = \pm\frac{3\pi}{2} \) within the discussed interval.
- The graph will never touch or cross these lines, emphasizing the limits imposed by asymptotes.
Tangent Function
The tangent function, \( \tan x \), plays a significant role in trigonometry. Unlike sine and cosine, the tangent function has its own unique features including a different type of periodicity and vertical asymptotes. For typical applications:
- The tangent function relates to the slope of a corresponding angle in a right triangle.
- Its range is from \(-\infty\) to \(+\infty\), but it is undefined at certain points due to vertical asymptotes.
- Is vertically stretched by a factor of 2, meaning each positive and negative slope is twice steeper than the standard tangent function.
Vertical Shift
A vertical shift refers to moving a graph up or down a specific number of units. It does not affect the period or shape of the function. Instead, it just repositions it vertically along the y-axis. In the function \( y = -1 + 2 \tan x \):
- The \(-1\) indicates a downward shift by 1 unit.
- This change means all y-values are decreased by 1, so the entire graph is moved downward accordingly.
