To determine which function has a [tex]\( y \)[/tex]-intercept at -1 and an amplitude of 2, let's analyze each option step by step.
### Amplitude
Amplitude refers to the maximum distance the function's value gets from its average value, usually the coefficient in front of the trigonometric function (sine or cosine).
### Y-intercept
The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex].
#### Option 1: [tex]\( f(x) = -\sin(x) - 1 \)[/tex]
- Amplitude: The coefficient of [tex]\(\sin(x)\)[/tex] is -1, so the amplitude is [tex]\(|-1| = 1\)[/tex].
- Y-intercept: Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -\sin(0) - 1 = 0 - 1 = -1
\][/tex]
This function has the correct [tex]\( y \)[/tex]-intercept, but the amplitude is incorrect.
#### Option 2: [tex]\( f(x) = -2\sin(x) - 1 \)[/tex]
- Amplitude: The coefficient of [tex]\(\sin(x)\)[/tex] is -2, so the amplitude is [tex]\(|-2| = 2\)[/tex].
- Y-intercept: Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -2\sin(0) - 1 = 0 - 1 = -1
\][/tex]
This function has both the correct [tex]\( y \)[/tex]-intercept and amplitude.
#### Option 3: [tex]\( f(x) = -\cos(x) \)[/tex]
- Amplitude: The coefficient of [tex]\(\cos(x)\)[/tex] is -1, so the amplitude is [tex]\(|-1| = 1\)[/tex].
- Y-intercept: Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -\cos(0) = -1
\][/tex]
This function does not have the correct [tex]\( y \)[/tex]-intercept or amplitude.
#### Option 4: [tex]\( f(x) = -2\cos(x) - 1 \)[/tex]
- Amplitude: The coefficient of [tex]\(\cos(x)\)[/tex] is -2, so the amplitude is [tex]\(|-2| = 2\)[/tex].
- Y-intercept: Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -2\cos(0) - 1 = -2 \times 1 - 1 = -2 - 1 = -3
\][/tex]
This function has the correct amplitude but an incorrect [tex]\( y \)[/tex]-intercept.
In conclusion, the functions that have a [tex]\( y \)[/tex]-intercept at -1 and an amplitude of 2 are:
- [tex]\( f(x) = -2\sin(x) - 1 \)[/tex]
The correct options are therefore [tex]\( \boxed{2} \)[/tex].
