Answer:
[tex]y=6\cdot \sin(x-0.5)-3[/tex]
Explanation:
For the equation:
[tex]y=a\cdot \sin(b(x-c))+d[/tex] ...
As [tex]|a|[/tex] increases, the wave’s amplitude increases.
As [tex]b[/tex] increases, the wave’s period (wavelength) decreases.
[tex]\text{period} = \dfrac{2\pi}{b}[/tex]
As [tex]c[/tex] increases, the wave shifts to the right. (horizontal/phase shift)
As [tex]d[/tex] increases, the wave shifts upwards. (vertical shift)
We can solve for the amplitude of the given sine function by finding half the difference of its minimum and maximum y-values.
[tex]A=\frac{1}{2}(y_2 - y_1)[/tex]
[tex]A=\frac{1}{2}(3 - (-9))[/tex]
[tex]A=\frac{1}2(3 + 9)[/tex]
[tex]A=\frac{1}2(12)[/tex]
[tex]A=6[/tex]
This means that in the above equation ([tex]y=a\cdot \sin(b(x-c))+d[/tex]),
[tex]a = A = 6[/tex].
We know that [tex]b=1[/tex] because the period is [tex]2\pi[/tex].
We know that [tex]c=0.5[/tex] because the wave is shifted 0.5 units to the right.
We can find the vertical shift ([tex]d[/tex]) by adding the amplitude to the minimum y-value to get the center y-value of the function.
[tex]d= -9 + 6[/tex]
[tex]d = -3[/tex]
Finally, we can put these variables together to form the equation:
[tex]\boxed{y=6\cdot \sin(x-0.5)-3}[/tex]
