**Find the Specified Quantity: Phase Shift** On this page, we will learn how to find the phase shift, which is the horizontal translation of a trigonometric function. In this example, we will determine the phase shift of the following cosine function: \[ y = 3 \cos\left(\frac{1}{4}x + \frac{\pi}{4}\right) + 5 \] **Explanation of the Horizontal Translation:** The phase shift of a function in the form \( y = a \cos(bx + c) + d \) is calculated using the formula: \[ \text{Phase Shift} = -\frac{c}{b} \] In our example, we have: - \( b = \frac{1}{4} \) - \( c = \frac{\pi}{4} \) Hence, the phase shift is: \[ \text{Phase Shift} = -\frac{\frac{\pi}{4}}{\frac{1}{4}} = -\pi \] Therefore, the function is horizontally translated by \(-\pi\) units. This means the graph of the cosine function shifts to the left by \(\pi\) units.
**Find the Specified Quantity: Phase Shift** On this page, we will learn how to find the phase shift, which is the horizontal translation of a trigonometric function. In this example, we will determine the phase shift of the following cosine function: \[ y = 3 \cos\left(\frac{1}{4}x + \frac{\pi}{4}\right) + 5 \] **Explanation of the Horizontal Translation:** The phase shift of a function in the form \( y = a \cos(bx + c) + d \) is calculated using the formula: \[ \text{Phase Shift} = -\frac{c}{b} \] In our example, we have: - \( b = \frac{1}{4} \) - \( c = \frac{\pi}{4} \) Hence, the phase shift is: \[ \text{Phase Shift} = -\frac{\frac{\pi}{4}}{\frac{1}{4}} = -\pi \] Therefore, the function is horizontally translated by \(-\pi\) units. This means the graph of the cosine function shifts to the left by \(\pi\) units.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![**Find the Specified Quantity: Phase Shift**
On this page, we will learn how to find the phase shift, which is the horizontal translation of a trigonometric function. In this example, we will determine the phase shift of the following cosine function:
\[ y = 3 \cos\left(\frac{1}{4}x + \frac{\pi}{4}\right) + 5 \]
**Explanation of the Horizontal Translation:**
The phase shift of a function in the form \( y = a \cos(bx + c) + d \) is calculated using the formula:
\[ \text{Phase Shift} = -\frac{c}{b} \]
In our example, we have:
- \( b = \frac{1}{4} \)
- \( c = \frac{\pi}{4} \)
Hence, the phase shift is:
\[ \text{Phase Shift} = -\frac{\frac{\pi}{4}}{\frac{1}{4}} = -\pi \]
Therefore, the function is horizontally translated by \(-\pi\) units. This means the graph of the cosine function shifts to the left by \(\pi\) units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb592224-c5dc-4690-b44a-e8ba296a38eb%2Fc2172145-071c-4a1a-b8df-e47e60091e60%2F8s60c8.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the Specified Quantity: Phase Shift**
On this page, we will learn how to find the phase shift, which is the horizontal translation of a trigonometric function. In this example, we will determine the phase shift of the following cosine function:
\[ y = 3 \cos\left(\frac{1}{4}x + \frac{\pi}{4}\right) + 5 \]
**Explanation of the Horizontal Translation:**
The phase shift of a function in the form \( y = a \cos(bx + c) + d \) is calculated using the formula:
\[ \text{Phase Shift} = -\frac{c}{b} \]
In our example, we have:
- \( b = \frac{1}{4} \)
- \( c = \frac{\pi}{4} \)
Hence, the phase shift is:
\[ \text{Phase Shift} = -\frac{\frac{\pi}{4}}{\frac{1}{4}} = -\pi \]
Therefore, the function is horizontally translated by \(-\pi\) units. This means the graph of the cosine function shifts to the left by \(\pi\) units.
Expert Solution
Step 1
For a periodic function of cosine: ., value c is called phase shift or the horizontal translation.
So, for the given function the value of phase shift is .
Step by step
Solved in 2 steps
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