Designer functions Design a sine function with the given properties. NKS It has a period o

Step 1
Find the sine function satisying the properties.
The sine function: ${\displaystyle f\left(x\right)=A\sin (\frac{2\pi }{p}\cdot (x-d))+c}$ has amplitude A, ${\displaystyle A=\frac{f_1-f_2}{2}}$ with the period p hr, d is the horizontal shift and c is the vertical shift. If ${\displaystyle d>0}$ graph shifts towards right and if ${\displaystyle d<0}$ graph shifts towards left, ${\displaystyle d=\frac{t_1+t_2}{2}}$. If ${\displaystyle c>0}$ graph shifts upward and if ${\displaystyle c<0}$ graph shifts downward, ${\displaystyle c=\frac{f_1+f_2}{2}}$.
Here ${\displaystyle f_1}$ and ${\displaystyle f_2}$ are the maximum and minimum value of function respectively, ${\displaystyle t_1}$ and ${\displaystyle t_2}$ are the points where function attains maximum and minimum value respectively.
Sine function with period of 12 hr with minimum value of 10 at ${\displaystyle t=3hr}$ and maximum value of 16 at ${\displaystyle t=15hr.}$
We have ${\displaystyle p=12,f_1=4,f_2=-4,t_1=3}$ and ${\displaystyle t_2=9}$
Step 2
Calculate A, d and c.
${\displaystyle A=\frac{4-(-4)}{2}}$
${\displaystyle =4}$
${\displaystyle d=\frac{3+9}{2}}$
${\displaystyle =6}$
and
${\displaystyle c=\frac{4-4}{2}}$
${\displaystyle =0}$
Substitute the values in the function:
${\displaystyle f\left(x\right)=4\sin (\frac{2\pi }{12}\cdot (x-6))+0}$
${\displaystyle =4\sin (\frac{\pi }{6}\cdot (x-6))}$