Answer :
Let's break down the given functions and identify their amplitude, period, and midline step-by-step.
We'll analyze each function one at a time.
### 1. [tex]\( 6 \sin (4x - 10) + 3 \)[/tex]
Amplitude: The amplitude is the coefficient of the sine function. Here, it is 6.
Period: The period of a sine or cosine function [tex]\( \sin(Bx + C) \)[/tex] or [tex]\( \cos(Bx + C) \)[/tex] is calculated as [tex]\( \frac{2\pi}{|B|} \)[/tex].
For this function, [tex]\( B = 4 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{4} = \frac{\pi}{2} \][/tex]
Midline: The midline is the constant term added at the end of the function. Here, it is 3.
So, for [tex]\( 6 \sin (4x - 10) + 3 \)[/tex]:
- Amplitude: [tex]\( 6 \)[/tex]
- Period: [tex]\( \frac{\pi}{2} \)[/tex]
- Midline: [tex]\( 3 \)[/tex]
### 2. [tex]\( \frac{1}{3} \sin (6x + 4) - 10 \)[/tex]
Amplitude: The amplitude is the coefficient of the sine function. Here, it is [tex]\( \frac{1}{3} \)[/tex].
Period: For this function, [tex]\( B = 6 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{6} = \frac{\pi}{3} \][/tex]
Midline: The midline is the constant term subtracted at the end of the function. Here, it is [tex]\( -10 \)[/tex].
So, for [tex]\( \frac{1}{3} \sin (6x + 4) - 10 \)[/tex]:
- Amplitude: [tex]\( \frac{1}{3} \)[/tex]
- Period: [tex]\( \frac{\pi}{3} \)[/tex]
- Midline: [tex]\( -10 \)[/tex]
### 3. [tex]\( \frac{2}{3} \cos (7x + 10) - 4 \)[/tex]
Amplitude: The amplitude is the coefficient of the cosine function. Here, it is [tex]\( \frac{2}{3} \)[/tex].
Period: For this function, [tex]\( B = 7 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{7} \][/tex]
Midline: The midline is the constant term subtracted at the end of the function. Here, it is [tex]\( -4 \)[/tex].
So, for [tex]\( \frac{2}{3} \cos (7x + 10) - 4 \)[/tex]:
- Amplitude: [tex]\( \frac{2}{3} \)[/tex]
- Period: [tex]\( \frac{2\pi}{7} \)[/tex]
- Midline: [tex]\( -4 \)[/tex]
### 4. [tex]\( -10 \cos \left(\frac{2}{3} x + \frac{1}{6}\right) + 8 \)[/tex]
Amplitude: The amplitude is the coefficient of the cosine function, ignoring the negative sign since amplitude is always positive. Here, it is [tex]\( 10 \)[/tex].
Period: For this function, [tex]\( B = \frac{2}{3} \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{\frac{2}{3}} = 3 \times 2\pi = 6\pi \approx 9.42477796076938 \][/tex]
Midline: The midline is the constant term added at the end of the function. Here, it is [tex]\( 8 \)[/tex].
So, for [tex]\( -10 \cos \left(\frac{2}{3} x + \frac{1}{6}\right) + 8 \)[/tex]:
- Amplitude: [tex]\( 10 \)[/tex]
- Period: [tex]\( 6\pi \)[/tex]
- Midline: [tex]\( 8 \)[/tex]
### Summary
| Function | Amplitude | Period | Midline |
|---------------------------------------------|-------------|-------------------|---------|
| [tex]\( 6 \sin (4x - 10) + 3 \)[/tex] | [tex]\( 6 \)[/tex] | [tex]\( \frac{\pi}{2} \)[/tex] | [tex]\( 3 \)[/tex] |
| [tex]\( \frac{1}{3} \sin (6x + 4) - 10 \)[/tex] | [tex]\( \frac{1}{3} \)[/tex] | [tex]\( \frac{\pi}{3} \)[/tex] | [tex]\( -10 \)[/tex] |
| [tex]\( \frac{2}{3} \cos (7x + 10) - 4 \)[/tex] | [tex]\( \frac{2}{3} \)[/tex] | [tex]\( \frac{2\pi}{7} \)[/tex] | [tex]\( -4 \)[/tex] |
| [tex]\( -10 \cos \left(\frac{2}{3} x + \frac{1}{6}\right) + 8 \)[/tex] | [tex]\( 10 \)[/tex] | [tex]\( 6\pi \approx 9.42477796076938 \)[/tex] | [tex]\( 8 \)[/tex] |
We'll analyze each function one at a time.
### 1. [tex]\( 6 \sin (4x - 10) + 3 \)[/tex]
Amplitude: The amplitude is the coefficient of the sine function. Here, it is 6.
Period: The period of a sine or cosine function [tex]\( \sin(Bx + C) \)[/tex] or [tex]\( \cos(Bx + C) \)[/tex] is calculated as [tex]\( \frac{2\pi}{|B|} \)[/tex].
For this function, [tex]\( B = 4 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{4} = \frac{\pi}{2} \][/tex]
Midline: The midline is the constant term added at the end of the function. Here, it is 3.
So, for [tex]\( 6 \sin (4x - 10) + 3 \)[/tex]:
- Amplitude: [tex]\( 6 \)[/tex]
- Period: [tex]\( \frac{\pi}{2} \)[/tex]
- Midline: [tex]\( 3 \)[/tex]
### 2. [tex]\( \frac{1}{3} \sin (6x + 4) - 10 \)[/tex]
Amplitude: The amplitude is the coefficient of the sine function. Here, it is [tex]\( \frac{1}{3} \)[/tex].
Period: For this function, [tex]\( B = 6 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{6} = \frac{\pi}{3} \][/tex]
Midline: The midline is the constant term subtracted at the end of the function. Here, it is [tex]\( -10 \)[/tex].
So, for [tex]\( \frac{1}{3} \sin (6x + 4) - 10 \)[/tex]:
- Amplitude: [tex]\( \frac{1}{3} \)[/tex]
- Period: [tex]\( \frac{\pi}{3} \)[/tex]
- Midline: [tex]\( -10 \)[/tex]
### 3. [tex]\( \frac{2}{3} \cos (7x + 10) - 4 \)[/tex]
Amplitude: The amplitude is the coefficient of the cosine function. Here, it is [tex]\( \frac{2}{3} \)[/tex].
Period: For this function, [tex]\( B = 7 \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{7} \][/tex]
Midline: The midline is the constant term subtracted at the end of the function. Here, it is [tex]\( -4 \)[/tex].
So, for [tex]\( \frac{2}{3} \cos (7x + 10) - 4 \)[/tex]:
- Amplitude: [tex]\( \frac{2}{3} \)[/tex]
- Period: [tex]\( \frac{2\pi}{7} \)[/tex]
- Midline: [tex]\( -4 \)[/tex]
### 4. [tex]\( -10 \cos \left(\frac{2}{3} x + \frac{1}{6}\right) + 8 \)[/tex]
Amplitude: The amplitude is the coefficient of the cosine function, ignoring the negative sign since amplitude is always positive. Here, it is [tex]\( 10 \)[/tex].
Period: For this function, [tex]\( B = \frac{2}{3} \)[/tex].
[tex]\[ \text{Period} = \frac{2\pi}{\frac{2}{3}} = 3 \times 2\pi = 6\pi \approx 9.42477796076938 \][/tex]
Midline: The midline is the constant term added at the end of the function. Here, it is [tex]\( 8 \)[/tex].
So, for [tex]\( -10 \cos \left(\frac{2}{3} x + \frac{1}{6}\right) + 8 \)[/tex]:
- Amplitude: [tex]\( 10 \)[/tex]
- Period: [tex]\( 6\pi \)[/tex]
- Midline: [tex]\( 8 \)[/tex]
### Summary
| Function | Amplitude | Period | Midline |
|---------------------------------------------|-------------|-------------------|---------|
| [tex]\( 6 \sin (4x - 10) + 3 \)[/tex] | [tex]\( 6 \)[/tex] | [tex]\( \frac{\pi}{2} \)[/tex] | [tex]\( 3 \)[/tex] |
| [tex]\( \frac{1}{3} \sin (6x + 4) - 10 \)[/tex] | [tex]\( \frac{1}{3} \)[/tex] | [tex]\( \frac{\pi}{3} \)[/tex] | [tex]\( -10 \)[/tex] |
| [tex]\( \frac{2}{3} \cos (7x + 10) - 4 \)[/tex] | [tex]\( \frac{2}{3} \)[/tex] | [tex]\( \frac{2\pi}{7} \)[/tex] | [tex]\( -4 \)[/tex] |
| [tex]\( -10 \cos \left(\frac{2}{3} x + \frac{1}{6}\right) + 8 \)[/tex] | [tex]\( 10 \)[/tex] | [tex]\( 6\pi \approx 9.42477796076938 \)[/tex] | [tex]\( 8 \)[/tex] |
