f (x) =-12 cos 2x+ +7, find the amplitude, period, horizontal shift 3
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
![The function given is:
\[ f(x) = -12 \cos \left( 2x + \frac{\pi}{3} \right) + 7 \]
To find the properties of the cosine function, we need to determine the amplitude, period, and horizontal shift.
### Amplitude:
The amplitude is determined by the coefficient of the cosine function. Here, the amplitude is \( |-12| = 12 \).
### Period:
The period of a cosine function is given by the formula:
\[ \text{Period} = \frac{2\pi}{b} \]
where \( b \) is the coefficient of \( x \) inside the cosine function. Here, \( b = 2 \), so the period is:
\[ \frac{2\pi}{2} = \pi \]
### Horizontal Shift:
The horizontal shift (or phase shift) is determined by the expression inside the cosine function. The formula used is:
\[ \text{Horizontal shift} = -\frac{c}{b} \]
where \( c \) is the constant added inside the cosine function. In this case, \( c = \frac{\pi}{3} \) and \( b = 2 \), so the horizontal shift is:
\[ -\frac{\pi/3}{2} = -\frac{\pi}{6} \]
This means the graph shifts \(\frac{\pi}{6}\) units to the left.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8492a80-fb1a-4bc0-8e68-f5fde27cb209%2F3abeca47-fab4-4cd5-b4d6-6e31371d8e14%2Fpabcssv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The function given is:
\[ f(x) = -12 \cos \left( 2x + \frac{\pi}{3} \right) + 7 \]
To find the properties of the cosine function, we need to determine the amplitude, period, and horizontal shift.
### Amplitude:
The amplitude is determined by the coefficient of the cosine function. Here, the amplitude is \( |-12| = 12 \).
### Period:
The period of a cosine function is given by the formula:
\[ \text{Period} = \frac{2\pi}{b} \]
where \( b \) is the coefficient of \( x \) inside the cosine function. Here, \( b = 2 \), so the period is:
\[ \frac{2\pi}{2} = \pi \]
### Horizontal Shift:
The horizontal shift (or phase shift) is determined by the expression inside the cosine function. The formula used is:
\[ \text{Horizontal shift} = -\frac{c}{b} \]
where \( c \) is the constant added inside the cosine function. In this case, \( c = \frac{\pi}{3} \) and \( b = 2 \), so the horizontal shift is:
\[ -\frac{\pi/3}{2} = -\frac{\pi}{6} \]
This means the graph shifts \(\frac{\pi}{6}\) units to the left.
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