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Magazine Chapter 5: Problem 6
Graph the function. $$f(x)=-1+\cos x$$
Short Answer
Expert verified
Graph \(f(x) = -1 + \cos x\) by shifting the cosine wave down by 1; it ranges from -2 to 0 and passes through key points like \((0, 0)\) and \((\pi, -2)\).
Step by step solution
01
Understanding the Function
The function given is \( f(x) = -1 + \cos x \). This is a cosine function that has been vertically shifted downward by 1 unit.
02
Identify Key Characteristics of \( \cos x \)
The basic cosine function, \( \cos x \), has a period of \( 2\pi \), an amplitude of 1, and oscillates between -1 and 1. The graph is symmetric about the y-axis.
03
Apply the Vertical Shift
The function \( f(x) = -1 + \cos x \) takes the graph of \( \cos x \) and shifts it downward by 1 unit. This means the new maximum is 0 and the new minimum is -2.
04
Determine the Range
With the vertical shift considered, the range of \( f(x) = -1 + \cos x \) is from \(-2\) to \(0\).
05
Find Critical Points
For one full period from \(0\) to \(2\pi\), the critical points of \(f(x)\) where \(\cos x\) typically takes values of 1, 0, and -1 are now at \(-1 + 1 = 0\), \(-1 + 0 = -1\), and \(-1 - 1 = -2\), respectively. Thus, these values occur at \((0, 0)\), \(\left(\frac{\pi}{2}, -1\right)\), \((\pi, -2)\), \(\left(\frac{3\pi}{2}, -1\right)\), and \((2\pi, 0)\).
06
Sketch the Graph
Plot the critical points on the coordinate system and draw a smooth curve passing through these points. The curve should start at \((0,0)\), dip to \((\pi,-2)\), rise back to \((2\pi,0)\), reflecting the cosine wave shifted down by 1 unit.
07
Label the Graph
Label the x-axis in terms of \(\pi\) for easy reading of key points. The y-axis should include the range from \(-2\) to \(0\), with tick marks at \(-1\) and \(-2\) for clarity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosine Function
The cosine function, represented as \( \cos x \), is one of the fundamental trigonometric functions. It is periodic, meaning it repeats its values in regular intervals or periods. The standard form of the cosine function has some distinct characteristics:
- The amplitude, which is the height from the midline to the peak, is 1. This means \( \cos x \) oscillates between 1 and -1.
- The period is \(2\pi\), indicating the distance along the x-axis before the function starts repeating its pattern.
- Cosine waves are symmetric around the y-axis, providing them a distinctive 'wave-like' appearance.
Vertical Shift
A vertical shift involves moving the entire graph of a function up or down on the Cartesian plane. When we say \( f(x) = -1 + \cos x \), the "-1" indicates a downward shift.
This affects every point on the graph of the function \(\cos x\). Here's how:
This affects every point on the graph of the function \(\cos x\). Here's how:
- Every y-value that the cosine function takes is reduced by 1.
- This shifts the maximum from 1 to 0 and the minimum from -1 to -2.
Function Range
The range of a function is the set of all possible y-values that the function can output. For the basic cosine function \( \cos x \), the range is between -1 and 1.
However, with the vertical shift included in \( f(x) = -1 + \cos x \), the range changes:
However, with the vertical shift included in \( f(x) = -1 + \cos x \), the range changes:
- Previously maximum becomes the new maximum at 0.
- Previously minimum is the new minimum at -2.
Critical Points
Critical points in a trigonometric function like the cosine function are where the curve changes direction 鈥� the peaks, troughs, and midline crossings.
For the basic cosine function \( \cos x \), these points typically occur where \( x \) values correspond to angles where cosine values are 1, 0, and -1. In the case of \( f(x) = -1 + \cos x \):
For the basic cosine function \( \cos x \), these points typically occur where \( x \) values correspond to angles where cosine values are 1, 0, and -1. In the case of \( f(x) = -1 + \cos x \):
- The maximum at 1 becomes 0, occurring at points such as \( (0,0) \) and \( (2\pi,0) \).
- The zeros at 0 become -1, occurring at points like \( \left( \frac{\pi}{2}, -1 \right) \) and \( \left( \frac{3\pi}{2}, -1 \right) \).
- The minimum at -1 becomes -2, occurring at the angle \( \pi \), i.e., \( (\pi, -2) \).
