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Chapter 4: Problem 99

Sketch the graph of \(y=\cos b x\) for \(b=\frac{1}{2}\) \(2,\) and \(3 .\) How does the value of \(b\) affect the graph? How many complete cycles of the graph of \(y\) occur between 0 and \(2 \pi\) for each value of \(b ?\)

Short Answer

Expert verified
The value of \(b\) determines the frequency of the cosine function. For \(b=\frac{1}{2}\), there is a half cycle between 0 and \(2 \pi\). For \(b=2\), there are two complete cycles, and for \(b=3\), there are three complete cycles within this range.

Step by step solution

01

Sketching the Graph for \(b=\frac{1}{2}\)

The function \(y=\cos(\frac{1}{2}x)\) features a horizontal stretch. This means the graph of cosine will take twice as long to complete a cycle. Sketch the general wave shape of cosine, but make sure that one complete wavelength extends from \(x=0\) to \(x=4\pi\) (instead of from 0 to \(2\pi\) as in the basic cosine function).
02

Sketching the Graph for \(b=2\)

The function \(y=\cos(2x)\) has a horizontal compression. This means it will go twice as quickly through its cycle. The wavelength here will be from 0 to \(\pi\), so we should sketch two cycles between 0 and \(2\pi\).
03

Sketching the Graph for \(b=3\)

In the function \(y=\cos(3x)\), we see an even faster movement through the cycle. Here the wavelength goes from 0 to \(\frac{2\pi}{3}\), thus in the range from 0 to \(2\pi\), we will observe 3 full cycles.
04

Discussing the Impact of Varying \(b\)

The value of \(b\) affects the frequency of the graph. A higher \(b\) results in a higher frequency, which means more cycles are completed within a given range. A smaller \(b\) results in a lower frequency, leading to fewer cycles in the same range.
05

Counting the Cycles Between 0 and \(2 \pi\)

For \(b=\frac{1}{2}\), there is one half of a cycle, for \(b=2\), there are two complete cycles, and for \(b=3\), there are three complete cycles between 0 and \(2 \pi\).

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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, denoted as \(y = \cos(x)\), is a trigonometric function that represents a smooth periodic wave. Its range is from -1 to 1, and in its basic form, completes one full cycle over the interval from 0 to \(2\pi\). The cosine wave is symmetrical, starting at its maximum point at \(x = 0\) and returning to this point after completing a cycle.

Key characteristics include:
  • Amplitude: The height from the centerline to the peak, typically 1 for \(\cos(x)\).
  • Period: The length of one complete cycle, which is \(2\pi\) for \(\cos(x)\).
  • Midline: The horizontal line that divides the wave into equal upper and lower sections, usually the x-axis for \(\cos(x)\).
Understanding these properties helps in graphing and modifying the cosine function when changes like frequency or phase shifts are introduced.
Frequency
Frequency is a crucial aspect when discussing trigonometric functions like the cosine function. It refers to how many cycles of the wave occur in a given interval. Specifically, for the function \(y = \cos(bx)\), the frequency is dictated by the value of \(b\). A higher frequency means that more cycles fit within a specific range, such as 0 to \(2\pi\).

Here's how it works:
  • Higher Frequency (\(b > 1\)): As \(b\) increases, the cosine wave oscillates more rapidly, squeezing more cycles into the same space. For instance, if \(b = 3\), three complete cycles appear between 0 and \(2\pi\).
  • Lower Frequency (\(b < 1\)): A smaller \(b\) slows down the oscillation, stretching the cycles further apart. For \(b = 0.5\), half a cycle fits within 0 to \(2\pi\).
Understanding frequency helps in predicting how many waves will appear on a graph for any given value of \(b\).
Wavelength
Wavelength is directly related to frequency and it tells us the distance over which the wave's shape repeats. For a cosine function \(y = \cos(bx)\), the wavelength can be calculated by the formula \(\frac{2\pi}{b}\).

Here鈥檚 what happens when the value of \(b\) changes:
  • Shorter Wavelength (\(b > 1\)): When \(b\) is greater than 1, the wavelength shortens, cramming more waves in a smaller span on the x-axis. For example, with \(b = 2\), the wavelength becomes \(\pi\), leading to two cycles from 0 to \(2\pi\).
  • Longer Wavelength (\(b < 1\)): Conversely, when \(b\) is less than 1, the wavelength extends, leaving more space between cycles. For \(b = 0.5\), the wavelength becomes \(4\pi\), so only half a cycle fits from 0 to \(2\pi\).
Understanding wavelength is essential for graphing and anticipating the stretching or compressing of the wave along the x-axis.
Horizontal Stretch/Compression
Horizontal stretch and compression affect the spacing of the cycles in the cosine graph. These transformations alter the period of the function \(y = \cos(bx)\) by stretching or compressing it horizontally based on the factor \(b\).

The effects on the graph include:
  • Horizontal Stretch (\(0 < b < 1\)): A smaller \(b\) causes the wave to stretch horizontally, lengthening the period. This results in fewer cycles over a fixed x-interval. For \(b = 0.5\), the period is \(4\pi\), stretching the graph.
  • Horizontal Compression (\(b > 1\)): A larger \(b\) compresses the wave, shortening the period. This results in more cycles appearing over the same x-interval. If \(b = 3\), the period is \(\frac{2\pi}{3}\), compressing the graph.
Recognizing how a graph stretches or compresses assists in visualizing and sketching accurate representations of trigonometric functions.

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Most popular questions from this chapter

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