The functions below represent transformations of the function f (x) = cos(x). Select the correct transformation represented. a) g (x) = cos(3x) Select] b) h (x) = cos(r) – 4 [ Select] c) k (x) = 4 cos(x) [Select]
The functions below represent transformations of the function f (x) = cos(x). Select the correct transformation represented. a) g (x) = cos(3x) Select] b) h (x) = cos(r) – 4 [ Select] c) k (x) = 4 cos(x) [Select]
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,...
Related questions
Question
![### Understanding the Geometry of a Circle and Trigonometric Relationships
For the circle defined by the equation \( x^2 + y^2 = r^2 \):
#### Circumference of the Circle:
\[ \text{Circumference} = 2\pi r \]
#### Trigonometric Relationships with Angle \( \theta \):
If \( \theta \) is an angle in standard position, and its terminal ray passes through the point \( P(x, y) \) on the circle, then the trigonometric functions are defined as follows:
\[ \sin \theta = \frac{y}{r} \]
\[ \cos \theta = \frac{x}{r} \]
\[ \tan \theta = \frac{y}{x} \]
Additionally, the Pythagorean identity holds:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
#### Conversion Between Degrees and Radians:
\[ 180 \text{ degrees} = \pi \text{ radians} \]
### Diagram Explanation
The diagram accompanying the text illustrates the circle with radius \( r \). It contains a coordinate system with horizontal and vertical axes intersecting at the center of the circle. A point \( P(x, y) \) is marked on the circumference of the circle. An angle \( \theta \) is shown in standard position with its vertex at the origin (the center of the circle) and its terminal side passing through point \( P(x, y) \).
The angle \( \theta \) and the radius \( r \) form a right triangle with the x-axis, where:
- \( x \) is the adjacent side,
- \( y \) is the opposite side,
- and \( r \) is the hypotenuse (radius of the circle).
This setup visually supports the trigonometric relationships defined above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F945c8a79-affc-4bd6-bac5-afc665d5b633%2F975e1ebd-8144-4785-956d-dbcac44b9a2d%2Fz9glswg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Geometry of a Circle and Trigonometric Relationships
For the circle defined by the equation \( x^2 + y^2 = r^2 \):
#### Circumference of the Circle:
\[ \text{Circumference} = 2\pi r \]
#### Trigonometric Relationships with Angle \( \theta \):
If \( \theta \) is an angle in standard position, and its terminal ray passes through the point \( P(x, y) \) on the circle, then the trigonometric functions are defined as follows:
\[ \sin \theta = \frac{y}{r} \]
\[ \cos \theta = \frac{x}{r} \]
\[ \tan \theta = \frac{y}{x} \]
Additionally, the Pythagorean identity holds:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
#### Conversion Between Degrees and Radians:
\[ 180 \text{ degrees} = \pi \text{ radians} \]
### Diagram Explanation
The diagram accompanying the text illustrates the circle with radius \( r \). It contains a coordinate system with horizontal and vertical axes intersecting at the center of the circle. A point \( P(x, y) \) is marked on the circumference of the circle. An angle \( \theta \) is shown in standard position with its vertex at the origin (the center of the circle) and its terminal side passing through point \( P(x, y) \).
The angle \( \theta \) and the radius \( r \) form a right triangle with the x-axis, where:
- \( x \) is the adjacent side,
- \( y \) is the opposite side,
- and \( r \) is the hypotenuse (radius of the circle).
This setup visually supports the trigonometric relationships defined above.
![The functions below represent transformations of the function \( f(x) = \cos(x) \). Select the correct transformation represented.
a) \( g(x) = \cos(3x) \)
[Select]
b) \( h(x) = \cos(x) - 4 \)
[Select]
c) \( k(x) = 4 \cos(x) \)
[Select]
*Note: This section should allow the word 'Select' to be interactive dropdown menus from which students can choose the correct transformation types. For instance, the choices could include options such as "Horizontal Stretch," "Vertical Shift Up/Down," "Amplitude Change," and so on, to help students understand the specific type of transformation applied to the function \( f(x) = \cos(x) \).*
In this context:
- \( g(x) = \cos(3x) \) represents a horizontal compression.
- \( h(x) = \cos(x) - 4 \) represents a vertical shift down by 4 units.
- \( k(x) = 4 \cos(x) \) represents a vertical stretch by a factor of 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F945c8a79-affc-4bd6-bac5-afc665d5b633%2F975e1ebd-8144-4785-956d-dbcac44b9a2d%2Fb5hzqjo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The functions below represent transformations of the function \( f(x) = \cos(x) \). Select the correct transformation represented.
a) \( g(x) = \cos(3x) \)
[Select]
b) \( h(x) = \cos(x) - 4 \)
[Select]
c) \( k(x) = 4 \cos(x) \)
[Select]
*Note: This section should allow the word 'Select' to be interactive dropdown menus from which students can choose the correct transformation types. For instance, the choices could include options such as "Horizontal Stretch," "Vertical Shift Up/Down," "Amplitude Change," and so on, to help students understand the specific type of transformation applied to the function \( f(x) = \cos(x) \).*
In this context:
- \( g(x) = \cos(3x) \) represents a horizontal compression.
- \( h(x) = \cos(x) - 4 \) represents a vertical shift down by 4 units.
- \( k(x) = 4 \cos(x) \) represents a vertical stretch by a factor of 4.
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 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education