For each of the following functions, state
- the domain;
- the range;
- the minimum and maximum values;
- the period;
- the phase shift;
- and the amplitude.
a) $f(x)=-3 \cos \left(\frac{1}{6} x-2 \pi\right)+4$
b) $f(x)=\frac{1}{2} \sin \left(2 x+\frac{\pi}{2}\right)-2$
Answer
\boxed{\text{Domain: } (-\infty, \infty), \text{ Range: } [1, 7], \text{ Min: } 1, \text{ Max: } 7, \text{ Period: } 12\pi, \text{ Phase shift: } 0, \text{ Amplitude: } 3}
Step 1 ：\(f(x) = -3 \cos \left(\frac{1}{6} x - 2 \pi \right) + 4\)
Step 2 ：\text{Domain: } (-\infty, \infty)
Step 3 ：\text{Range: } [1, 7]
Step 4 ：\text{Minimum value: } 1
Step 5 ：\text{Maximum value: } 7
Step 6 ：\text{Period: } 12\pi
Step 7 ：\text{Phase shift: } 0
Step 8 ：\text{Amplitude: } 3
Step 9 ：\boxed{\text{Domain: } (-\infty, \infty), \text{ Range: } [1, 7], \text{ Min: } 1, \text{ Max: } 7, \text{ Period: } 12\pi, \text{ Phase shift: } 0, \text{ Amplitude: } 3}
