Question

what is the range of $y = -5\sin(x)$?
\bigcirc all real numbers $-5 \leq y \leq 5$
\bigcirc all real numbers $-\frac{5}{2} \leq y \leq \frac{5}{2}$
\bigcirc all real numbers $-1 \leq y \leq 1$
\bigcirc all real numbers $-\frac{1}{5} \leq y \leq \frac{1}{5}$

Explanation:

Step1: Recall the range of \( \sin(x) \)

The range of the sine function \( \sin(x) \) is \( -1 \leq \sin(x) \leq 1 \).

Step2: Apply the vertical stretch and reflection

For the function \( y = -5\sin(x) \), we multiply \( \sin(x) \) by \( -5 \). Multiplying by a negative number reflects the graph over the x - axis, but the range is determined by the absolute value of the coefficient. The range of \( a\sin(x) \) (where \( a\) is a constant) is \( -|a| \leq y \leq |a| \). Here, \( |a|=| - 5| = 5 \), so the range of \( y=-5\sin(x) \) is \( - 5\leq y\leq5 \).

Answer:

all real numbers \( -5\leq y\leq5 \)