Problem 10 In Exercises 9-24, sketch the gr... [FREE SOLUTION]

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Trigonometry

Cynthia Y. Young

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 4: Problem 10

In Exercises 9-24, sketch the graph of each sinusoidal function over one period. $$ y=3-\cos x $$

Short Answer

Expert verified

Graph the key points: $(0, 2), (\pi/2, 3), (\pi, 4), (3\pi/2, 3), (2\pi, 2)$. Connect to sketch the sinusoidal wave over one period.

Step by step solution

Understand the Basic Cosine Function

The basic cosine function has the equation: $ y = \cos x $. Its graph has a period of $ 2\pi $, an amplitude of 1, and it starts at the maximum value of 1 when $ x = 0 $.

Adjust the Amplitude and Vertical Shift

For the function $ y = 3 - \cos x $, the amplitude (value in front of the cosine function) is 1, since there is no coefficient directly in front of $ \cos x $. However, there's a vertical shift of +3. This means every point on the cosine curve is shifted up by 3 units.

Determine the Key Points

For one period, the basic cosine function $ \cos x $ has key points at $ (0, 1), (\pi/2, 0), (\pi, -1), (3\pi/2, 0), (2\pi, 1) $. With the vertical shift, these key points for $ y = 3 - \cos x $ become: $(0, 3-1), (\pi/2, 3-0), (\pi, 3+1), (3\pi/2, 3-0), (2\pi, 3-1)$. Simplifying, these are $(0, 2), (\pi/2, 3), (\pi, 4), (3\pi/2, 3), (2\pi, 2)$.

Sketch the Graph

Now plot these key points on a graph. Starting from $ (0, 2) $, the curve will increase to $ (\pi/2, 3) $, reach its maximum at $ (\pi, 4) $, then decrease back to $ (3\pi/2, 3) $, and finally return to $ (2\pi, 2) $. This will complete one full period of the sinusoidal function.

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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 is a fundamental trigonometric function often represented as $ y = \cos x $. It is periodic, which means it repeats its pattern over regular intervals. The typical graph of the cosine function has a period of $ 2\pi $, covering the x-values from 0 to $ 2\pi $ for one complete cycle.

The graph of $ y = \cos x $ starts at its maximum value of 1, when $ x = 0 $.
It moves down to 0 at $ x = \pi/2 $, reaches its minimum at $ x = \pi $ where $ y = -1 $, and goes back to 0 at $ x = 3\pi/2 $.
Finally, it completes the cycle at $ x = 2\pi $, returning to the maximum value.

Understanding the basic cosine function is essential for modifying it to suit different equation requirements, such as changing amplitude or applying shifts.

Amplitude

Amplitude refers to the distance from the middle of the wave to its peak or trough. For the basic cosine function $ y = \cos x $, the amplitude is 1, as the values fluctuate between -1 and 1.
In equations of the form $ y = a \cdot \cos x $, "a" represents the amplitude. The amplitude affects how "tall" the wave appears. For instance:

If $ a = 2 $, then the graph goes from -2 to 2, making the wave twice as tall.
If $ a = 0.5 $, the values fluctuate from -0.5 to 0.5, resulting in a more "compressed" wave.

In our specific case, $ y = 3 - \cos x $, the amplitude remains 1 since there is no coefficient other than 1 in front of $ \cos x $. This means the height of the typical peaks and troughs remains unchanged, but the vertical shift changes their position.

Vertical Shift

A vertical shift involves moving a graph up or down on the coordinate plane without altering its shape. In the function $ y = 3 - \cos x $, there is no scaling of the $ \cos x $ term, indicated by no multiplier before it. The graph is shifted vertically upwards by 3 units due to the "3" in front of the function.

A vertical shift does not affect the function's period or amplitude, but rather its position on the y-axis.
This means that each point on the basic cosine graph moves up by three units.
For instance, if a cosine function originally peaked at $ y = 1 $, it will peak at $ y = 4 $ after this shift.

Applying a vertical shift alters the key values but retains the overall structure and periodic nature of the cosine wave.

Graph Sketching

Graph sketching is the process of visually representing a function to understand its behavior over one period, or more. For drawing sinusoidal functions like $ y = 3 - \cos x $, sketching helps visualize the transformation done by shifts and scaling.

Start by identifying any shifts: in our exercise, it's a vertical shift of +3.
Plot key points based on the shift. For example: $ (0,2), (\pi/2,3), (\pi,4), (3\pi/2,3), (2\pi,2) $.
Draw a smooth, continuous curve linking these points, ensuring it follows the upward and downward nature of cosines.

These points and curves collectively outline one complete period of the function, providing a visual insight into the function鈥檚 fluctuations and transformations.

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91影视

In Exercises 9-24, sketch the graph of each sinusoidal function over one period. $$ y=3-\cos x $$

Short Answer

Step by step solution

Understand the Basic Cosine Function

Adjust the Amplitude and Vertical Shift

Determine the Key Points

Sketch the Graph

Key Concepts

Cosine Function

Amplitude

Vertical Shift

Graph Sketching

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