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Magazine Chapter 4: Problem 70
Graph one complete cycle for each of the following. In each case, label the axes accurately and state the period, vertical translation, and horizontal translation for each graph. \(y=1+\tan \left(2 x-\frac{\pi}{4}\right)\)
Short Answer
Expert verified
The period is \( \frac{\pi}{2} \), there's a vertical shift up 1, and a horizontal shift right by \( \frac{\pi}{8} \).
Step by step solution
01
Understanding the Function
The given function is \( y = 1 + \tan\left(2x - \frac{\pi}{4}\right) \). This is a transformation of the basic tangent function \( y = \tan(x) \). We need to identify the key transformations: vertical translation, horizontal translation, and the period change.
02
Determine Vertical Translation
The number \( +1 \) outside the tangent function indicates a vertical translation. Therefore, the whole graph is shifted 1 unit upwards.
03
Calculate Period of the Tangent Function
The period of \( y = \tan(x) \) is \( \pi \). For \( y = \tan(bx) \), the period becomes \( \frac{\pi}{b} \). In our function, \( b = 2 \), so the period is \( \frac{\pi}{2} \).
04
Identify Horizontal Translation
The expression inside the tangent function is \( 2x - \frac{\pi}{4} \), which indicates a horizontal translation. Solve \( 2x - \frac{\pi}{4} = 0 \) to find the translation. By solving, \( x = \frac{\pi}{8} \), the graph is shifted \( \frac{\pi}{8} \) units to the right.
05
Graphing the Function
Plot the basic tangent cycle with updated characteristics. Start one complete cycle from \( x = \frac{\pi}{8} \) to \( x = \frac{\pi}{8} + \frac{\pi}{2} = \frac{5\pi}{8} \). Mark the vertical asymptotes and the point where the curve crosses the vertical translated line at \( y = 1 \).
06
Label the Axes
Label the x-axis with values starting from \( x = \frac{\pi}{8} \) to \( x = \frac{5\pi}{8} \). The y-axis should be labeled with increments around \( y = 1 \), the vertical translation line. The asymptotes occur at \( \frac{3\pi}{16} \) to \( \frac{7\pi}{16} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tangent Function
The tangent function, often denoted as \( y = \tan(x) \), is one of the fundamental trigonometric functions. It relates to the sine and cosine functions through the ratio \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This means that wherever the cosine of an angle is zero, the tangent function experiences a discontinuity and goes to infinity, resulting in vertical asymptotes.
- **Behavior**: Unlike sine and cosine, tangent has a period of \( \pi \), emphasizing more frequent cycles.
- **Graph Characteristics**: The graph of the tangent function is a series of repeating "S" shaped curves.
- **Asymptotic Nature**: It has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \).
Graph Transformations
Graph transformations involve altering the basic ways in which a function's graph appears on the Cartesian plane. These changes can include translations, reflections, and dilations. For the tangent function transformation:
These transformations modify where the graph begins and end within its cycle, but not its inherent shape.
- **Vertical Translation**: Adding a constant outside the function, such as \(+1\) in \( y = 1 + \tan(x) \), raises or lowers the graph by that amount. In our original problem, this moves the entire graph up by 1 unit.
- **Horizontal Translation**: This involves a shift left or right. It occurs from manipulating the variable inside the function. For instance, in \( y = \tan(2x - \frac{\pi}{4}) \), the inside change causes a right shift by \( \frac{\pi}{8} \) units.
These transformations modify where the graph begins and end within its cycle, but not its inherent shape.
Period of a Function
The period of a function indicates the distance it takes to complete one full cycle of its pattern. For the tangent function \( y = \tan(x) \), the natural period is \( \pi \). However, transformations can alter this:
In our transformed tangent function example of \( y = \tan(2x - \frac{\pi}{4}) \), the "2" introduces a stretch factor dividing the standard period by two, resulting in a period of \( \frac{\pi}{2} \). This reflects a quicker cycle repetition, where the graph completes one cycle between \( x = \frac{\pi}{8} \) and \( x = \frac{5\pi}{8} \).
- **Scaling**: For a tangent function expressed as \( y = \tan(bx) \), where \( b \) is a constant, the period becomes \( \frac{\pi}{b} \). This means the function repeats its pattern more frequently if \( b > 1 \), or less frequently if \( b < 1 \).
In our transformed tangent function example of \( y = \tan(2x - \frac{\pi}{4}) \), the "2" introduces a stretch factor dividing the standard period by two, resulting in a period of \( \frac{\pi}{2} \). This reflects a quicker cycle repetition, where the graph completes one cycle between \( x = \frac{\pi}{8} \) and \( x = \frac{5\pi}{8} \).
Asymptotes
Asymptotes are lines that a curve approaches but never actually touches. They are critical in understanding the behavior of the tangent function:
Understanding the position and nature of asymptotes helps in accurately sketching graph behaviors and predicting points of discontinuity.
- **Vertical Asymptotes**: These occur where the tangent function is undefined, typically at odd multiples of \( \frac{\pi}{2} \). This is where the denominator (cosine) equals zero.
- **Locations**: In any transformed tangent equation, such as \( y = \tan(2x - \frac{\pi}{4}) \), asymptotes are vital in graphing. Here, the vertical asymptotes are calculated within the new cycle, occurring at points like \( x = \frac{3\pi}{16} \) and \( x = \frac{7\pi}{16} \).
Understanding the position and nature of asymptotes helps in accurately sketching graph behaviors and predicting points of discontinuity.
