Question

The graph of a sinusoidal function has a minimum point at (0, - 10) and then has a maximum point at (2, - 4) Write the formula of the function , where x is entered in radians .

Answer 1

Answer:   $\bold{y=3sin\bigg(\dfrac{\pi}{2}x-\dfrac{\pi}{2}\bigg)-7}$

Step-by-step explanation:

Minimum: (0, -10)

Maximum: (2, -4)

y = A sin (Bx - C) + D

• Amplitude (A) = (Max - Min)/2
• Period = 2π/B   →   B = 2π/Period
• Phase Shift = C/B     →    C = B × Phase Shift
• Midline (D) = (Max + Min)/2

$A=\dfrac{-4-(-10)}{2}\quad =\dfrac{-4+10}{2}\quad =\dfrac{6}{2}\quad =\large\boxed{3}$

$\text{x-value of Max minus x-value of Min}= \dfrac{1}{2}\text{Period}\\\\2 - 0 = \dfrac{1}{2}P\quad \rightarrow \quad P=4\\\\\\B=\dfrac{2\pi}{P}\quad =\dfrac{2\pi}{4}\quad =\large\boxed{\dfrac{\pi}{2}}$

$D = \dfrac{\text{Max + Min}}{2}\quad = \dfrac{-4+(-10)}{2}\quad =\dfrac{-14}{2}\quad =\large\boxed{-7}$

Sin usually starts at (0, 0).  For this graph, the midline touches 0 when x = 1 so the Phase Shift = 1.

$C = B \times \text{Phase Shift}\quad = \dfrac{\pi}{2}\times 1\quad =\large\boxed{\dfrac{\pi}{2}}$

$A=3, \quad B=\dfrac{\pi}{2}, \quad C=\dfrac{\pi}{2},\quad D=-7\\\\\rightarrow \quad \large\boxed{y=3\sin \bigg(\dfrac{\pi}{2}x-\dfrac{\pi}{2}\bigg)-7}$