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Chapter 4: Problem 99

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

Short Answer

Expert verified
The graph of the function \(y=|x| \sin x\) is an oscillating graph that increases in amplitude as x moves away from the origin.

Step by step solution

01

Understanding the |x| function

The absolute value function \(|x|\) is a function that makes all input positive. If x is positive, it remains positive. If it is negative, it turns positive. The graph of such a function generally is a V-shaped curve.
02

Understanding the sin(x) function

The sine function \(\sin x\) is a periodic function that oscillates between -1 and 1. The period of the sine function is \(2\pi\), which means it repeats its pattern every \(2\pi\) units.
03

Interpreting the |x|sin(x) function

Multiplying the absolute value function \(|x|\) and the sine function \(\sin x\) combines the properties of both. Since \(\sin x\) oscillates between -1 and 1, the resulting function \(|x|\sin x\) will be zero at x = 0 and will also oscillate, but its amplitude increases as x moves away from the origin because of the |x| factor.
04

Graphing the function

To successfully create a graph of the function \(y=|x| \sin x\), plot several points. For example, plotting points where x= integer multiples of \(\pi\) and \(-\pi\), the function should exhibit an oscillatory behaviour. The graph can then be created by connecting these points in a smooth manner.

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Key Concepts

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

Absolute Value
The absolute value function, denoted as \(|x|\), transforms any input number into its non-negative counterpart. This means:
  • If \(x\) is positive, \(|x| = x\).
  • If \(x\) is negative, \(|x| = -x\).
This leads to the characteristic V-shaped graph. The point of the V is located at the origin (0,0), where the graph changes direction. Understanding this transformation helps visualize how the function behaves across different parts of the graph, particularly when combined with other functions like sine.
Sine Function
The sine function, represented as \(\sin x\), is known for its wave-like pattern. It oscillates smoothly between -1 and 1 and has a period of \(2\pi\). This means the shape of its graph repeats every \(2\pi\) units.
  • The peak value of the wave is 1, while the trough is -1.
  • Key points where \(\sin x = 0\) include multiples of \(\pi\), like \(0, \pi, 2\pi\), etc.
These oscillations are essential in understanding how sine interacts with other functions, especially because it adds periodic components to any graph it's involved with.
Graphing Techniques
Graphing a function like \(y = |x| \sin x\) involves combining the behaviors of both \(|x|\) and \(\sin x\). Here's a quick approach:
  • Start by identifying key points from both \(|x|\) and \(\sin x\).
  • Consider key multiples of \(\pi\) since they often mark zeros and turning points.
  • As \(x\) increases, note that the amplitude of sine increases due to \(|x|\).
To draw this graph, plot these combined effects:
  • Where \(x = 0\), the function is zero.
  • On either side of zero, the graph rises and falls, reflecting the influence of \(\sin x\) while being stretched by \(|x|\).
For a smooth graph, connect the plotted points with a flowing curve, capturing both the oscillation and growth.
Function Analysis
Analyzing the function \(y = |x| \sin x\) involves examining the effects of both components:
  • At \(x = 0\), the value is zero because \(|0| = 0\).
  • For positive \(x\), the function behaves like \(x \sin x\), gradually increasing in amplitude.
  • For negative \(x\), it mirrors the positive side because the absolute value makes \(x\) positive.
Observing this function, notice:
  • Its oscillatory pattern similar to sine, but each wave's height adjusts according to \(|x|\).
  • The symmetry around the y-axis, thanks to the absolute value ensuring \(y\) is always non-negative.
This kind of analysis helps predict how the graph behaves even without plotting every single point.

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