Which function will have a \( y \)-intercept at -1 and an amplitude of 2 ? \( f(x)=-\sin (x)-1 \) \( f(x)=-2 \sin (x)-1 \) \( f(x)=-\cos (x) \) \( f(x)=-2 \cos (x)-1 \)

To determine which function has a \( y \)-intercept at -1 and an amplitude of 2, we need to analyze each function based on these criteria. 1. **Amplitude**: The amplitude of a sine or cosine function \( A \sin(x) \) or \( A \cos(x) \) is given by the absolute value of the coefficient \( A \). Therefore, the amplitude is \( |A| \). 2. **Y-intercept**: The \( y \)-intercept of a function \( f(x) \) is found by evaluating \( f(0) \). Now, let's analyze each function: 1. **Function**: \( f(x) = -\sin(x) - 1 \) - Amplitude: \( |A| = |-1| = 1 \) - Y-intercept: \( f(0) = -\sin(0) - 1 = -0 - 1 = -1 \) 2. **Function**: \( f(x) = -2 \sin(x) - 1 \) - Amplitude: \( |A| = |-2| = 2 \) - Y-intercept: \( f(0) = -2 \sin(0) - 1 = -2(0) - 1 = -1 \) 3. **Function**: \( f(x) = -\cos(x) \) - Amplitude: \( |A| = |-1| = 1 \) - Y-intercept: \( f(0) = -\cos(0) = -1 \) 4. **Function**: \( f(x) = -2 \cos(x) - 1 \) - Amplitude: \( |A| = |-2| = 2 \) - Y-intercept: \( f(0) = -2 \cos(0) - 1 = -2(1) - 1 = -2 - 1 = -3 \) Now, summarizing the results: - \( f(x) = -\sin(x) - 1 \): Amplitude = 1, Y-intercept = -1 - \( f(x) = -2 \sin(x) - 1 \): Amplitude = 2, Y-intercept = -1 - \( f(x) = -\cos(x) \): Amplitude = 1, Y-intercept = -1 - \( f(x) = -2 \cos(x) - 1 \): Amplitude = 2, Y-intercept = -3 The function that has a \( y \)-intercept at -1 and an amplitude of 2 is: \[ \boxed{f(x) = -2 \sin(x) - 1} \]