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Chapter 6: Problem 82

Find the maximum and minimum values of the function. $$y=\frac{1}{2} \sin (2 x-6 \pi)-4$$

Short Answer

Expert verified
The minimum value is -4.5, and the maximum value is -3.5.

Step by step solution

01

- Understand the given function

The given function is \(y = \frac{1}{2} \sin (2x - 6\pi) - 4\). Notice it's a transformation of the basic sine function.
02

- Identify the amplitude of the sine function

The amplitude of the sine function \( \sin(kx + b) \) is given by the coefficient in front of the sine term. Here, the amplitude is \( \frac{1}{2} \).
03

- Determine the range of the sine function

The sine function \( \sin(2x - 6\pi) \) oscillates between -1 and 1. When multiplied by \( \frac{1}{2} \), the range becomes \[ -\frac{1}{2} \leq \frac{1}{2}\sin(2x - 6\pi) \leq \frac{1}{2} \].
04

- Translate the range based on the function

The function is translated downwards by 4 units. Therefore, translate the range: \( \frac{1}{2} \sin (2x - 6\pi) - 4 \) leads to \[ -\frac{1}{2} - 4 \leq y \leq \frac{1}{2} - 4 \].
05

- Simplify the range

Simplifying the translated range: \[ -\frac{1}{2} - 4 = -4.5 \] and \[ \frac{1}{2} - 4 = -3.5 \]. Hence, we have \[ -4.5 \leq y \leq -3.5 \].

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Key Concepts

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

sine function transformation
The given function is a transformation of the basic sine function. A transformation means changes are applied to the basic pattern of the sine wave. For the function \(y = \frac{1}{2} \sin (2x - 6\pi) - 4\), different transformations occur simultaneously. These include changes in amplitude, horizontal phase shift, and vertical translation.
  • The coefficient \(2\) inside the sine function argument affects the period of the function.
  • The term \(-6\pi\) shifts the sine wave horizontally.
  • The factor \(\frac{1}{2}\) changes the amplitude of the wave.
  • The term \(-4\) translates the function vertically downwards.
Each transformation changes the appearance and values the sine function can take.
amplitude
Amplitude is the height of the wave from its centerline to its peak. In the function \(y = \frac{1}{2} \sin (2x - 6\pi) - 4\), the amplitude can be found by looking at the coefficient in front of the sine function.
Here, the amplitude is \(\frac{1}{2}\). This means the wave will rise up to \(\frac{1}{2}\) unit above and fall \(\frac{1}{2}\) unit below the centerline before any vertical shift is applied.
In a real-world context, amplitude can represent how strong or loud a signal is, like with sound waves or electrical signals.
function range
The range of a function is the set of all possible output values. For the sine function, the basic range is from \(-1\) to \(1\). However, for our transformed function \(y = \frac{1}{2} \sin (2x - 6\pi) - 4\), we need to consider the transformations.
First, the amplitude transformation changes the range to \(-\frac{1}{2}\) to \(\frac{1}{2}\).
Next, the vertical translation shifts this range.
Since the function is translated downward by 4 units, the new range becomes \(-4.5\) to \(-3.5\).
vertical translation
Vertical translation shifts the entire graph of the function up or down. This is achieved by adding or subtracting a constant from the function.
In the function \(y = \frac{1}{2} \sin (2x - 6\pi) - 4\), the term \(-4\) means the graph is shifted 4 units downwards.
This translation affects the range by moving all output values downward without changing the amplitude.
Therefore, if the original sine function ranged from \(-\frac{1}{2}\) to \(\frac{1}{2}\), after vertical translation, the range updates to \(-4.5\) to \(-3.5\).

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