Problem 68 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 4: Problem 68

Use a graphing utility to graph each function. $$y=\sin |x|$$

Short Answer

Expert verified

The graph of the function $y = \sin |x|$ is obtained by graphing $y = \sin x$ for $x > 0$, and creating a mirror image of this graph for $x < 0$ because of the absolute value. The resulting graph is symmetrical about the y-axis and oscillates between -1 and 1.

Step by step solution

Understanding the Function

The function given is $y = \sin |x|$. It is a sinusoidal function but with an absolute value applied to $x$. The $\sin$ function oscillates between -1 and 1 as $x$ increases or decreases. However, the absolute value operator will reflect the part of the function for $x < 0$ to the positive side making its graph symmetrical about the y-axis.

Graphing the Positive Side

For $x > 0$, the function $y = \sin |x|$ behaves exactly as $y = \sin x$. So, graph $y = \sin x$ for $x > 0$.

Reflecting for Negative x

For $x < 0$, $|x|= -x$, hence $y= \sin(-x)$. But, $\sin(-x) = -\sin(x)$, which is the reflection of the $\sin$ function in x-axis. However, due to absolute value operator, instead of flipping in x-axis, it flips in y-axis. Therefore, for $x < 0$ the graph of $y = \sin |x|$ is a reflection about the y-axis of the graph for $x > 0$.

Joining the Symmetrical Graphs

Combine the steps to derive the graph for the function $y = \sin |x|$. The graph starts at the origin (0,0). For $x > 0$, it follows the usual shape of $y = \sin x$. For $x < 0$, it is the mirror image of the graph for $x > 0$. The final function graph is therefore symmetrical about the y-axis.

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Key Concepts

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

Absolute Value Functions

Understanding absolute value functions is key to graphing many types of equations, including trigonometric ones such as $y = \sin |x|$. When you encounter an absolute value function, the primary role of the absolute value operator is to make negative inputs positive.
For example, $|x|$ equals $x$ when $x \geq 0$, but it changes to $-x$ when $x < 0$. In the context of trigonometric functions, this means the negative side of the input range is handled differently, leading to symmetry in the graph. The result is a function that is uniform across both positive and negative x-values.

Absolute value functions ensure that all reduced inputs are non-negative.
The key effect on graphs is creating a mirror image effect across the y-axis.
Understanding how absolute values impact transformation will help in sketching accurate graphs.

Symmetry in Graphs

Symmetry is a crucial concept, especially when graphing functions like $y = \sin |x|$. With symmetry, parts of the graph repeat themselves in a predictable way. For $y = \sin |x|$, symmetry is recognized about the y-axis. This is because:$ \sin |x| $ reflects at the y-axis, meaning whatever occurs on the positive x-axis will be mirrored on the negative x-axis.

When you graph these functions, understanding their symmetry helps reduce the effort. If you comprehend one side of the graph, the other is simply its mirror image.

Symmetry in graphs allows for transferring information from one side to the other.
This concept reduces complexity and aids in quicker graphing.
Recognizing symmetry can simplify calculations and visualizations.

Trigonometric Transformations

Trigonometric transformations play a significant role in how we see graphs like $y = \sin |x|$. While the base function $y = \sin(x)$ oscillates naturally between -1 and 1, the transformation applied by $|x|$ modifies its graphing.

Such transformations include shifts, stretches, and reflections. With $y = \sin |x|$, we see a reflection caused by the absolute value. This type of transformation maintains the amplitude and period of the original $\sin$ function but changes the way negative inputs affect the output.

Trigonometric transformations modify the traditional paths of sine and cosine curves.
These transformations can encompass reflections, translations, and stretches or compressions.
Understanding these modifications is essential for accurately sketching complex wave patterns.

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91影视

Use a graphing utility to graph each function. $$y=\sin |x|$$

Short Answer

Step by step solution

Understanding the Function

Graphing the Positive Side

Reflecting for Negative x

Joining the Symmetrical Graphs

Key Concepts

Absolute Value Functions

Symmetry in Graphs

Trigonometric Transformations

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