What is the range of \( y=-3 \sin (x)-4 \) ? all real numbers \( -7 \leq y \leq 7 \) all real numbers \( -7 \leq y \leq-1 \) all real numbers \( -5 \leq y \leq 3 \) all real numbers \( -1 \leq y \leq 1 \)

To find the range of the function \( y = -3 \sin(x) - 4 \), let's first consider the range of \( \sin(x) \), which is between -1 and 1. Now, substituting that into the equation we get: 1. When \( \sin(x) = 1 \): \[ y = -3(1) - 4 = -3 - 4 = -7 \] 2. When \( \sin(x) = -1 \): \[ y = -3(-1) - 4 = 3 - 4 = -1 \] Thus, the range of \( y \) is from -7 to -1, which translates to all real numbers between -7 and -1, inclusive. So, the correct answer is: all real numbers \( -7 \leq y \leq -1 \). Now, for a bit more fun and engaging knowledge: The sine function is fascinating and has been studied for hundreds of years! Ancient Greeks even explored its properties, which laid the groundwork for trigonometry as we know it today. So, when you are analyzing functions involving sine, you're tapping into a deep historical context of mathematics! If you ever find yourself working with trigonometric functions, be careful with the amplitude and vertical shifts! It's easy to mistake the transformations. A common mistake is not flipping the sign when including a negative amplitude, which can lead you astray from identifying the correct range. Always sketch it out for clarity!