# Designer functions Design a sine function with the given properties. NSK It has

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

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Step 1
Find the sine function satisying the properties.
The sine function: $$\displaystyle{f{{\left({x}\right)}}}={A}{\sin{{\left({\frac{{{2}\pi}}{{{p}}}}\cdot{\left({x}-{d}\right)}\right)}}}+{c}$$ 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 $$\displaystyle{t}={3}{h}{r}$$ and maximum value of 16 at $$\displaystyle{t}={15}{h}{r}.$$
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.
$$\displaystyle{A}={\frac{{{4}-{\left(-{4}\right)}}}{{{2}}}}$$
$$\displaystyle={4}$$
$$\displaystyle{d}={\frac{{{3}+{9}}}{{{2}}}}$$
$$\displaystyle={6}$$
and
$$\displaystyle{c}={\frac{{{4}-{4}}}{{{2}}}}$$
$$\displaystyle={0}$$
Substitute the values in the function:
$$\displaystyle{f{{\left({x}\right)}}}={4}{\sin{{\left({\frac{{{2}\pi}}{{{12}}}}\cdot{\left({x}-{6}\right)}\right)}}}+{0}$$
$$\displaystyle={4}{\sin{{\left({\frac{{\pi}}{{{6}}}}\cdot{\left({x}-{6}\right)}\right)}}}$$