What is the range of [tex]$y=-3 \sin (x)-4$[/tex]?

The sine function sin ( x ) ranges from -1 to 1.
To find the minimum value of y , set sin ( x ) = 1 : y min ​ = − 3 ( 1 ) − 4 = − 7 .
To find the maximum value of y , set sin ( x ) = − 1 : y ma x ​ = − 3 ( − 1 ) − 4 = − 1 .
The range of y is − 7 ≤ y ≤ − 1 ​ .

Explanation

Understanding the Problem We are asked to find the range of the function y = − 3 sin ( x ) − 4 . We know that the sine function, sin ( x ) , always has values between -1 and 1, inclusive. That is, − 1 ≤ sin ( x ) ≤ 1 . We can use this fact to determine the range of the given function.

Finding Minimum and Maximum Values To find the minimum value of y , we need to maximize sin ( x ) , so we set sin ( x ) = 1 . Then, y min ​ = − 3 ( 1 ) − 4 = − 3 − 4 = − 7. To find the maximum value of y , we need to minimize sin ( x ) , so we set sin ( x ) = − 1 . Then, y ma x ​ = − 3 ( − 1 ) − 4 = 3 − 4 = − 1. Therefore, the range of y is − 7 ≤ y ≤ − 1 .

Determining the Range The range of the function y = − 3 sin ( x ) − 4 is all real numbers between -7 and -1, inclusive.


Examples
Understanding the range of trigonometric functions like y = − 3 sin ( x ) − 4 is useful in many real-world applications. For example, in electrical engineering, sinusoidal functions model alternating current (AC). If y represents the voltage of an AC circuit, knowing the range of y tells us the minimum and maximum voltage values, which is crucial for designing and protecting electrical components. Similarly, in acoustics, sinusoidal functions model sound waves, and the range of the function can represent the minimum and maximum sound pressure levels.

Answered by GinnyAnswer | 2025-07-03

The range of the function y = − 3 sin ( x ) − 4 is from -7 to -1, inclusive. This can be determined by using the known range of the sine function, which is from -1 to 1. By substituting these values into the function, we find the corresponding minimum and maximum values for y.
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Answered by Anonymous | 2025-07-04