Which function has the largest amplitude?

Question

(1 point)
Responses

f left parenthesis x right parenthesis equals negative 3 sine left parenthesis x minus Start Fraction pi over 4 End Fraction right parenthesis plus 1
Image with alt text: f left parenthesis x right parenthesis equals negative 3 sine left parenthesis x minus Start Fraction pi over 4 End Fraction right parenthesis plus 1

f left parenthesis x right parenthesis equals 2 cosine x plus 3
Image with alt text: f left parenthesis x right parenthesis equals 2 cosine x plus 3

f left parenthesis x right parenthesis equals sine left parenthesis 5 x plus Start Fraction 2 pi over 3 End Fraction right parenthesis minus 4
Image with alt text: f left parenthesis x right parenthesis equals sine left parenthesis 5 x plus Start Fraction 2 pi over 3 End Fraction right parenthesis minus 4

f left parenthesis x right parenthesis equals negative 2 cosine left parenthesis 4 x plus 3 pi right parenthesis minus 6

Answer 1

To find which function has the largest amplitude, we need to examine the amplitude of each of the given functions.

1. For the function \( f(x) = -3 \sin(x - \frac{\pi}{4}) + 1 \):

   • The amplitude of a sine function \( a \sin(bx + c) + d \) is given by \( |a| \).
   • Here, \( a = -3 \), so the amplitude is \( |-3| = 3 \).

2. For the function \( f(x) = 2 \cos(x) + 3 \):

   • Similarly, for this cosine function, the amplitude is given by \( |a| \).
   • Here, \( a = 2 \), so the amplitude is \( |2| = 2 \).

3. For the function \( f(x) = \sin(5x + \frac{2\pi}{3}) - 4 \):

   • Again, the amplitude is given by \( |a| \).
   • Here, \( a = 1 \), so the amplitude is \( |1| = 1 \).

4. For the function \( f(x) = -2\cos(4x + 3\pi) - 6 \):

   • The amplitude is given by \( |a| \) as well.
   • Here, \( a = -2 \), so the amplitude is \( |-2| = 2 \).

Now, let’s summarize the amplitudes:

• First function: Amplitude = 3
• Second function: Amplitude = 2
• Third function: Amplitude = 1
• Fourth function: Amplitude = 2

The function with the largest amplitude is the first function \( f(x) = -3 \sin(x - \frac{\pi}{4}) + 1 \) with an amplitude of 3.