Consider the following graph of the function f(z) which is a transformation of the cosine function. Find a formula for f(x). (You may assume that the period and any phase shift here are some nice multiple of . For example, or or 2, etc. The period or phase shift will not be something like 3 or 6.) f(x) = 0
Consider the following graph of the function f(z) which is a transformation of the cosine function. Find a formula for f(x). (You may assume that the period and any phase shift here are some nice multiple of . For example, or or 2, etc. The period or phase shift will not be something like 3 or 6.) f(x) = 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.5: Trigonometric Graphs
Problem 25E
Related questions
Question
![### Transformation of the Cosine Function
Consider the following graph of the function \( f(x) \) which is a transformation of the cosine function. Find a formula for \( f(x) \).
(You may assume that the period and any phase shift here are some nice multiple of π. For example, \( \frac{\pi}{2} \) or π or 2π, etc. The period or phase shift will **not** be something like 3 or 6.)
\[ f(x) = \]
#### Description of Graph:
The graph provided is a sinusoidal wave, which resembles a cosine function but has undergone transformations such as scaling and horizontal shifts. The x-axis is divided into \(\pi\) units, and the wave completes its period exactly at a multiple of \(\pi\).
The crucial points to note are:
- The peaks and troughs of the wave.
- The period of the wave, which can be derived from the distance between consecutive peaks or troughs.
- The amplitude (the maximum and minimum values) should be considered to identify any vertical scaling.
From visual inspection:
- The period appears to be \( \pi \).
- The amplitude (the vertical distance from the x-axis to the peak) seems to be 2.
- There is no horizontal shift observed directly in the snapshot.
Using these observations, we can construct a function \( f(x) \):
\[ f(x) = 2 \cos(2x) \]
Here, the cosine function is transformed with an amplitude of 2 (vertical scaling) and a period adjustment to \( \pi \) (indicated by the factor 2 inside the cosine function).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2F56ec0e82-ccf9-44e2-bfaa-c4b0ba22f12c%2Fmbvvmhc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transformation of the Cosine Function
Consider the following graph of the function \( f(x) \) which is a transformation of the cosine function. Find a formula for \( f(x) \).
(You may assume that the period and any phase shift here are some nice multiple of π. For example, \( \frac{\pi}{2} \) or π or 2π, etc. The period or phase shift will **not** be something like 3 or 6.)
\[ f(x) = \]
#### Description of Graph:
The graph provided is a sinusoidal wave, which resembles a cosine function but has undergone transformations such as scaling and horizontal shifts. The x-axis is divided into \(\pi\) units, and the wave completes its period exactly at a multiple of \(\pi\).
The crucial points to note are:
- The peaks and troughs of the wave.
- The period of the wave, which can be derived from the distance between consecutive peaks or troughs.
- The amplitude (the maximum and minimum values) should be considered to identify any vertical scaling.
From visual inspection:
- The period appears to be \( \pi \).
- The amplitude (the vertical distance from the x-axis to the peak) seems to be 2.
- There is no horizontal shift observed directly in the snapshot.
Using these observations, we can construct a function \( f(x) \):
\[ f(x) = 2 \cos(2x) \]
Here, the cosine function is transformed with an amplitude of 2 (vertical scaling) and a period adjustment to \( \pi \) (indicated by the factor 2 inside the cosine function).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage