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

Designer functions Design a sine function with the given properties. NKS It has a period of 12 hr with a minimum value of $-4$ at $t=0$ hr and a maximum value of 4 at $t=6$ hr.
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Nathaniel Kramer

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