Can the maximum and minimum for $y=5\sin(5x+20)-2$ be found using algebra?
Solution 1:
The $\sin t$ function takes values in $[-1,1]$. It takes the value $1$ when $t=\frac{\pi}{2}+2 k\pi, \,\,k \in \mathbb{Z}$ and it takes the value $-1$ when $t = -\frac{\pi}{2}+2k\pi, \,\,k \in \mathbb{Z}$. So, you function attains a maximum value of $3$ when $5x+20 =\frac{\pi}{2}+2 k\pi, \,\,k \in \mathbb{Z}$, and a minimum value of $-7$ when $5x+20 =-\frac{\pi}{2}+2 k\pi, \,\,k \in \mathbb{Z}$.
Solution 2:
Of course it can. Note that $-1 \le \sin(5x+20) \le 1$ and these bounds are achievable.