Answer: [tex]\bold{y=3sin\bigg(\dfrac{\pi}{2}x-\dfrac{\pi}{2}\bigg)-7}[/tex]
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
[tex]A=\dfrac{-4-(-10)}{2}\quad =\dfrac{-4+10}{2}\quad =\dfrac{6}{2}\quad =\large\boxed{3}[/tex]
[tex]\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}}\\[/tex]
[tex]D = \dfrac{\text{Max + Min}}{2}\quad = \dfrac{-4+(-10)}{2}\quad =\dfrac{-14}{2}\quad =\large\boxed{-7}[/tex]
Sin usually starts at (0, 0). For this graph, the midline touches 0 when x = 1 so the Phase Shift = 1.
[tex]C = B \times \text{Phase Shift}\quad = \dfrac{\pi}{2}\times 1\quad =\large\boxed{\dfrac{\pi}{2}}[/tex]
[tex]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}[/tex]
