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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solve for the equation. rewrite the equation in standard form. include the key x values of this equation. explain how you set of the graph for the function and include how the graph is moving. ( a example is included in one of the picture.
![**Show What You Know: Graphing Trigonometric Functions**
*MAT 190 - Precalculus*
---
### Objectives
The purpose of this assignment is for you to:
1. Demonstrate your ability to graph a trigonometric function;
2. Improve your mathematical writing to include full solutions with justifications;
3. Integrate mathematical statements into grammatically correct expositions.
---
### Recall Standard Form
Let \( A \), \( P \), \( H \), and \( V \) be real numbers with \( A \neq 0 \) and \( P > 0 \). Then,
\[ y = A \, \text{trig}(P(x - H)) + V, \]
where "trig" is the trigonometric function indicated by “sin”, “cos”, “csc”, “sec”, “cot”, or “tan”.
---
### Assignment
Draw a well-labeled graph of a single cycle of the trigonometric function
\[ f(x) = 3 \sin \left( \frac{\pi}{4} x \right) - 2. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c60962f-c8d5-4255-8d11-ba7fb1b92564%2F1c3e8df7-7d7a-4f1e-8611-a3565c2997c0%2Frquwtp7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Show What You Know: Graphing Trigonometric Functions**
*MAT 190 - Precalculus*
---
### Objectives
The purpose of this assignment is for you to:
1. Demonstrate your ability to graph a trigonometric function;
2. Improve your mathematical writing to include full solutions with justifications;
3. Integrate mathematical statements into grammatically correct expositions.
---
### Recall Standard Form
Let \( A \), \( P \), \( H \), and \( V \) be real numbers with \( A \neq 0 \) and \( P > 0 \). Then,
\[ y = A \, \text{trig}(P(x - H)) + V, \]
where "trig" is the trigonometric function indicated by “sin”, “cos”, “csc”, “sec”, “cot”, or “tan”.
---
### Assignment
Draw a well-labeled graph of a single cycle of the trigonometric function
\[ f(x) = 3 \sin \left( \frac{\pi}{4} x \right) - 2. \]
![To determine the three key \( y \)-values of the cosine function \( g(x) \), we utilize \( A \) and \( V \). The \( V = 2 \) tells us that the center line is shifted vertically up 2 units. The \( A = 1 \) tells us that the amplitude of the cosine function \( g(x) \) is \( |A| = 1 \), so we go up and down 1 unit from the center line to get the other two key \( y \)-values of \( V + |A| = 2 + 1 = 3 \) and \( V - |A| = 2 - 1 = 1 \).
To determine the five key \( x \)-values of the cosine function \( g(x) \), we utilize \( P \) and \( H \). The \( H = \frac{1}{2} \) tells us that we shift the cosine function \( g(x) \) to the right \(\frac{1}{2}\) unit to start a cycle. Since the period is
\[
\frac{2\pi}{P} = 2\pi - \frac{2}{\pi} = 4,
\]
the end of the cycle would be \( H + \frac{2\pi}{P} = \frac{1}{2} + 4 = \frac{9}{2} \). Between the key \( x \)-values of \(\frac{1}{2} \) and \(\frac{9}{2} \), the remaining three key \( x \)-values are evenly spaced, breaking that interval into four equal parts. Dividing the period by 4, we get \(\frac{4}{4} = 1\) as the distance between the key \( x \)-values. So, the remaining three key \( x \)-values are \(\frac{1}{2} + 1 = \frac{3}{2} \), \(\frac{3}{2} + 1 = \frac{5}{2} \), and \(\frac{5}{2} + 1 = \frac{7}{2} \).
With this information determined, we begin to graph our secant function \( f(x) \) by labeling the \( y \)-axis with the three key \( y \)-values and the \( x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c60962f-c8d5-4255-8d11-ba7fb1b92564%2F1c3e8df7-7d7a-4f1e-8611-a3565c2997c0%2Fyot2bjk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To determine the three key \( y \)-values of the cosine function \( g(x) \), we utilize \( A \) and \( V \). The \( V = 2 \) tells us that the center line is shifted vertically up 2 units. The \( A = 1 \) tells us that the amplitude of the cosine function \( g(x) \) is \( |A| = 1 \), so we go up and down 1 unit from the center line to get the other two key \( y \)-values of \( V + |A| = 2 + 1 = 3 \) and \( V - |A| = 2 - 1 = 1 \).
To determine the five key \( x \)-values of the cosine function \( g(x) \), we utilize \( P \) and \( H \). The \( H = \frac{1}{2} \) tells us that we shift the cosine function \( g(x) \) to the right \(\frac{1}{2}\) unit to start a cycle. Since the period is
\[
\frac{2\pi}{P} = 2\pi - \frac{2}{\pi} = 4,
\]
the end of the cycle would be \( H + \frac{2\pi}{P} = \frac{1}{2} + 4 = \frac{9}{2} \). Between the key \( x \)-values of \(\frac{1}{2} \) and \(\frac{9}{2} \), the remaining three key \( x \)-values are evenly spaced, breaking that interval into four equal parts. Dividing the period by 4, we get \(\frac{4}{4} = 1\) as the distance between the key \( x \)-values. So, the remaining three key \( x \)-values are \(\frac{1}{2} + 1 = \frac{3}{2} \), \(\frac{3}{2} + 1 = \frac{5}{2} \), and \(\frac{5}{2} + 1 = \frac{7}{2} \).
With this information determined, we begin to graph our secant function \( f(x) \) by labeling the \( y \)-axis with the three key \( y \)-values and the \( x
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