Problem 31 $23-36$ . Graph the function b... [FREE SOLUTION]

Chapter 1: Problem 31

$23-36$ . Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $$y=\frac{1}{2}(1-\cos x)$$

Short Answer

Expert verified

Start with $ \cos x $, reflect, shift up, and compress vertically.

Step by step solution

Identify the Standard Function

The given function is $ y = \frac{1}{2}(1 - \cos x) $. Recognize that the standard function here is $ \cos x $.

Transform the Standard Function

The transformation involved is modifying $ \cos x $ with $ 1 - \cos x $. This shifts the cosine graph upwards by 1 unit and reflects it across the x-axis.

Apply Vertical Compression

The factor $ \frac{1}{2} $ in $ \frac{1}{2}(1 - \cos x) $ compresses the graph vertically. This reduces the amplitude, making the range [0, 1].

Combine Transformations

Combine the transformations: the original $ \cos x $ graph is reflected over the x-axis, shifted up by 1, and then compressed vertically by factor of $ \frac{1}{2} $. The maximum value becomes 1 and the minimum becomes 0.

Final Graph

Begin with $ y = \cos x $, apply the transformations to follow the behavior of $ y = \frac{1}{2}(1 - \cos x) $. Sketch the curve from $ x = 0 $, passing through turning points at $(\pi, 0)$, $(2\pi, 1)$, forming a wave-like shape that repeats every $2\pi$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions

Trigonometric functions play a crucial role in mathematics, especially when it comes to modeling periodic phenomena. The primary trigonometric functions include sine ($ \sin x $ ), cosine ($ \cos x $ ), and tangent ($ \tan x $ ). These functions have specific patterns or waves, which makes them ideal for representing cyclical behaviors such as sound waves or circadian rhythms.

Sine and cosine functions are periodic with a period of $2\pi$.
The amplitude is the height from the center line to the peak.
Cosine starts at its maximum value when $ x = 0 $, while sine starts at zero.

Understanding these basic attributes of trigonometric functions is essential. They set the foundation for more complex transformations. For instance, we can modify the amplitude, period, and vertical shift to model different real-world situations.

Cosine Transformation

Transformations of the cosine function involve altering its basic shape or position. In the case of the function $ y = \frac{1}{2}(1 - \cos x) $, we are making adjustments on two fronts. First, consider the transformation inside the brackets: $1 - \cos x$. This shifts the graph up by 1 unit and effectively flips it around the x-axis. Originally, $\cos x$ starts at its peak, but $1 - \cos x$ starts at its lowest.By shifting and reflecting,

The peak of the cosine, usually at 1, starts now at 0.
The typical increase from 0 up to 1 becomes a decrease from 1 to 0.

These adjustments result in a reflection over the x-axis besides a shift, which is important when graphing as it changes both the vertical orientation and position of the graph.

Vertical Compression

Vertical compression changes the height of a wave graph without altering its horizontal dimensions. In the equation $ y = \frac{1}{2}(1 - \cos x) $, $\frac{1}{2}$ compresses the graph vertically, reducing the amplitude. This means the wave does not reach as high or dip as low as the original cosine wave.

The standard $ \cos x $ function has an amplitude of 1.
When compressed by a factor of $ \frac{1}{2} $, its maximum and minimum values become 0.5 and 0.
This makes the range from [0, 1] in the final form after other transformations have been applied.

It's vital to note that vertical compression affects the perceived intensity of fluctuations in data. Though it might look similar in shape, the graph is much squatter, emphasizing moderate changes over sharp peaks and troughs.

Reflection Over X-axis

A reflection of a function across the x-axis involves flipping the graph upside down. This transformation dramatically impacts how the graph behaves visually. In our case with $ y = \frac{1}{2}(1 - \cos x) $, the reflection is due to subtracting the cosine part from 1.

Typical peaks of cosine graphs get inverted.
When think of reflection, consider it as turning the peaks into troughs and vice versa.

In this case, the reflection happens before applying the other transformations, such as vertical compression or upward shifting. This approach is fundamental to ensuring the correct transformation sequence, leading to a wave-like curve that follows new peaks and troughs orientations. Understanding these transformations enables you to precisely predict the graph's appearance and behavior.

91影视

\(23-36\) . Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $$y=\frac{1}{2}(1-\cos x)$$

Short Answer

Step by step solution

Identify the Standard Function

Transform the Standard Function

Apply Vertical Compression

Combine Transformations

Final Graph

Key Concepts

Trigonometric Functions

Cosine Transformation

Vertical Compression

Reflection Over X-axis

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