# A minimum value of a sinusoidal Function is at (pi/4, 3). The nearest maximum value to the right of this point is at ((7pi)/12, 7). What is the equation of this function?

Dec 30, 2016

$y \left(x\right) = 2 \sin \left(3 x - \frac{5 \pi}{4}\right) + 5$

#### Explanation:

Let us write the generic sinusoidal function as:

$y \left(x\right) = A \sin \left(\alpha x + \beta\right) + B$

First we note that the value for the minimum is ${y}_{m} = 3$ while the maximum is for ${y}_{M} = 7$. From this we may derive:

$- A + B = 3$
$A + B = 7$

and therefore:

$A = 2$ and $B = 5$

Then we note that a minimum of $\sin x$ occurs for $x = - \frac{\pi}{2}$ and the next maximum for $x = \frac{\pi}{2}$, so we can derive:

$\alpha {x}_{m} + \beta = - \frac{\pi}{2}$
$\alpha {x}_{M} + \beta = \frac{\pi}{2}$

or:

$\alpha \frac{\pi}{4} + \beta = - \frac{\pi}{2}$

$\alpha \frac{7 \pi}{12} + \beta = \frac{\pi}{2}$

subtracting the first equation from the second:

$\alpha \left(\frac{7 \pi}{12} - \frac{\pi}{4}\right) = \pi$

so

$\alpha = \frac{1}{\frac{7}{12} - \frac{1}{4}} = \frac{1}{\frac{7}{12} - \frac{3}{12}} = \frac{12}{4} = 3$

and

$\beta = - \frac{\pi}{2} - \frac{3}{4} \pi = - \frac{5}{4} \pi$

Thus the sinusoidal function we are searching is:

$y \left(x\right) = 2 \sin \left(3 x - \frac{5 \pi}{4}\right) + 5$

graph{2sin(3x-(5pi)/4)+5 [-2.396, 2.604, 2, 8]}