To determine the range of the function [tex]\( y = -5 \sin(x) \)[/tex], let's examine the range of [tex]\(\sin(x)\)[/tex] and how it affects our function.
1. Understanding the range of [tex]\(\sin(x)\)[/tex]:
- The sine function, [tex]\(\sin(x)\)[/tex], oscillates between -1 and 1 for all real [tex]\(x\)[/tex].
- Therefore, the range of [tex]\(\sin(x)\)[/tex] is [tex]\([-1, 1]\)[/tex].
2. Applying the transformation:
- Our function is [tex]\( y = -5 \sin(x) \)[/tex].
- Multiplying [tex]\(\sin(x)\)[/tex] by -5 scales its range by a factor of -5.
3. Scaling the range:
- If [tex]\( \sin(x) = 1 \)[/tex], then [tex]\( y = -5 \cdot 1 = -5 \)[/tex].
- If [tex]\( \sin(x) = -1 \)[/tex], then [tex]\( y = -5 \cdot -1 = 5 \)[/tex].
- Consequently, the multiplication by -5 will invert and scale the range of [tex]\(\sin(x)\)[/tex], which means that all intermediate values are scaled similarly.
4. Finding the new range:
- When [tex]\(\sin(x)\)[/tex] is at its minimum value (-1), [tex]\(y\)[/tex] is at its maximum value (5).
- When [tex]\(\sin(x)\)[/tex] is at its maximum value (1), [tex]\(y\)[/tex] is at its minimum value (-5).
- Thus, the complete range of [tex]\( y = -5 \sin(x) \)[/tex] spans from -5 to 5, inclusive.
Therefore, the range of the function [tex]\( y = -5 \sin(x) \)[/tex] is all real numbers [tex]\( -5 \leq y \leq 5 \)[/tex].
The correct answer is:
[tex]$\text{all real numbers } -5 \leq y \leq 5 \$[/tex]
