Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Equation and Graph Analysis of Trigonometric Functions**
**Task:**
Write an equation of the form \( y = a \sin(bx) \) or \( y = a \cos(bx) \) to describe the graph below.
**Graph Description:**
The graph in the image is a trigonometric waveform, appearing to oscillate between positive and negative values. The x-axis is marked with increments of \( \frac{\pi}{4} \) and \( \frac{\pi}{8} \), up to \( 2\pi \). The y-axis ranges between -3 and 3.
**Detailed Explanation of the Graph:**
1. **Amplitude (a):**
- The maximum value of the graph is 2, and the minimum value is -2, so the amplitude \( a \) is \( 2 \).
2. **Period (T) and Frequency (b):**
- The period \( T \) of the graph is the length of one complete cycle.
- From the graph, one complete cycle occurs from 0 to \( \pi \), suggesting a period of \( \pi \).
- The period \( T \) is given by \( T = \frac{2\pi}{b} \). Hence, \( b \) can be calculated as:
\[
\pi = \frac{2\pi}{b} \implies b = 2
\]
3. **Function Type:**
- By inspecting the intercepts and the nature of the wave, we can identify whether it is a sine or cosine function.
- The graph starts at \( y = 0 \) when \( x = 0 \), which implies a sine function is likely, as the sine function \( \sin(0) = 0 \).
4. **Resulting Equation:**
- Given the amplitude, period, and type of function, the resulting equation can be written as:
\[
y = 2 \sin(2x)
\]
**Conclusion:**
Thus, the equation that describes the given graph is \( y = 2 \sin(2x) \).
_Figure:_
The figure above shows a trigonometric sine wave with specified amplitude, period, and intercepts on both the x-axis and y-axis.
---
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Transcribed Image Text:**Equation and Graph Analysis of Trigonometric Functions**
**Task:**
Write an equation of the form \( y = a \sin(bx) \) or \( y = a \cos(bx) \) to describe the graph below.
**Graph Description:**
The graph in the image is a trigonometric waveform, appearing to oscillate between positive and negative values. The x-axis is marked with increments of \( \frac{\pi}{4} \) and \( \frac{\pi}{8} \), up to \( 2\pi \). The y-axis ranges between -3 and 3.
**Detailed Explanation of the Graph:**
1. **Amplitude (a):**
- The maximum value of the graph is 2, and the minimum value is -2, so the amplitude \( a \) is \( 2 \).
2. **Period (T) and Frequency (b):**
- The period \( T \) of the graph is the length of one complete cycle.
- From the graph, one complete cycle occurs from 0 to \( \pi \), suggesting a period of \( \pi \).
- The period \( T \) is given by \( T = \frac{2\pi}{b} \). Hence, \( b \) can be calculated as:
\[
\pi = \frac{2\pi}{b} \implies b = 2
\]
3. **Function Type:**
- By inspecting the intercepts and the nature of the wave, we can identify whether it is a sine or cosine function.
- The graph starts at \( y = 0 \) when \( x = 0 \), which implies a sine function is likely, as the sine function \( \sin(0) = 0 \).
4. **Resulting Equation:**
- Given the amplitude, period, and type of function, the resulting equation can be written as:
\[
y = 2 \sin(2x)
\]
**Conclusion:**
Thus, the equation that describes the given graph is \( y = 2 \sin(2x) \).
_Figure:_
The figure above shows a trigonometric sine wave with specified amplitude, period, and intercepts on both the x-axis and y-axis.
---
**Continue Button:**
Navigate to the next step by selecting the "Continue" button at the
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