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Magazine Chapter 1: Problem 38
Sketch the graph of \(f,\) making use of stretching, reflecting, or shifting. (a) \(f(x)=\frac{1}{4} \tan x\) (b) \(f(x)=\tan (x+3 \pi / 4)\)
Short Answer
Expert verified
(a) Compress vertically, (b) Shift left by \( \frac{3\pi}{4} \).
Step by step solution
01
Understand the Tan Function
The basic tangent function, \( \tan(x) \), has a period of \( \pi \) and asymptotes at odd multiples of \( \frac{\pi}{2} \). It is an odd function, meaning it is symmetric about the origin. This will help you understand how transformations affect the graph.
02
Stretching \( \frac{1}{4} \tan x \)
The function \( f(x) = \frac{1}{4} \tan x \) indicates a vertical compression by a factor of 4. To plot this, take each point on the basic \( \tan(x) \) curve and scale its y-value by \( \frac{1}{4} \). The period and asymptotes remain the same.
03
Understand the Horizontal Shift \( \tan (x + \frac{3\pi}{4}) \)
The function \( f(x) = \tan(x + \frac{3\pi}{4}) \) implies a horizontal shift. The \( +\frac{3\pi}{4} \) shifts the basic tangent curve to the left by \( \frac{3\pi}{4} \). The asymptotes also shift to the left by this amount, but the period remains \( \pi \).
04
Sketching \( f(x) = \frac{1}{4} \tan x \) Graph
Draw the basic framework of \( \tan(x) \) with its usual asymptotes at \( x = \frac{\pi}{2} + k\pi \). Vertically compress this entire graph by a factor of 4, so that each y-value becomes smaller by that factor. The graph stays between the same asymptotes.
05
Sketching \( f(x) = \tan(x + \frac{3\pi}{4}) \) Graph
Begin with the basic \( \tan(x) \) graph. Shift the entire graph left by \( \frac{3\pi}{4} \) units. Remember to shift all asymptotes by \( \frac{3\pi}{4} \) units in the negative x-direction as well.
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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, a fundamental trigonometric function, is known for its unique wave pattern.
The basic form of this function, denoted as \( \tan(x) \), has key characteristics you should grasp:
The basic form of this function, denoted as \( \tan(x) \), has key characteristics you should grasp:
- Periodicity: The tangent function repeats every \( \pi \) radians, meaning it has a period of \( \pi \).
- Asymptotes: It has vertical asymptotes at odd multiples of \( \frac{\pi}{2} \), such as \( \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}, \text{etc.} \).
- Symmetry: This is an odd function, so it's symmetric about the origin. If you reflect its graph across both the x- and y-axes, it remains unchanged.
Horizontal Shift
Horizontal shifting affects the position of the graph along the x-axis.
When you see an expression like \( \tan(x + \frac{3\pi}{4}) \), it tells you that the entire graph of the tangent function is shifting left or right.
In this case, the \( +\frac{3\pi}{4} \) indicates a shift to the left by \( \frac{3\pi}{4} \) units.
When you see an expression like \( \tan(x + \frac{3\pi}{4}) \), it tells you that the entire graph of the tangent function is shifting left or right.
In this case, the \( +\frac{3\pi}{4} \) indicates a shift to the left by \( \frac{3\pi}{4} \) units.
- Graph Movement: Each point on the graph, along with the asymptotes, moves to the left by this amount.
- Period Preservation: Despite this shift, the period of the function remains \( \pi \).
Vertical Compression
Vertical compression alters how steeply a graph rises or falls.
In the function \( \frac{1}{4} \tan(x) \), the factor \( \frac{1}{4} \) causes a vertical compression.
This means that each point on the tangent curve is scaled towards the x-axis:
In the function \( \frac{1}{4} \tan(x) \), the factor \( \frac{1}{4} \) causes a vertical compression.
This means that each point on the tangent curve is scaled towards the x-axis:
- Amplitude Change: The graph's y-values are reduced to a quarter of their original size, making it flatter.
- No Period Change: The period and position of the asymptotes do not change through compression.
Sketching Graphs
Sketching graphs requires a step-by-step approach for clarity.
When sketching \( \frac{1}{4} \tan(x) \), begin by sketching the basic \( \tan(x) \) curve.
When sketching \( \frac{1}{4} \tan(x) \), begin by sketching the basic \( \tan(x) \) curve.
- Identify the asymptotes at \( x = \frac{\pi}{2} + k\pi \).
- Apply the vertical compression, scaling all y-values down by \( \frac{1}{4} \).
- Plot the basic \( \tan(x) \).
- Shift the graph left by \( \frac{3\pi}{4} \), including all asymptotes and central features.
