Answer:
Step-by-step explanation:
Midline is the horizontal line that passes exactly in the middle between the graph's maximum and minimum points.
Amplitude is the vertical distance between the midline and one of the extremum points.
Period is the distance between two consecutive maximum points, or two consecutive minimum points (these distances must be equal).
To find the formula for [tex]g(x)[/tex]. First, let's use the given information to determine the function's amplitude, midline, and period.
The midline of a sinusoidal function passes exactly in the middle of its extreme values. So we can find the midline by finding the average of the maximum and minimum values.
The midline passes exactly between the minimum value 1, and the maximum value 11, so the midline equation is
[tex]midline=y=\frac{11+1}{2} =6[/tex]
The extremum points are 5 units above or below the midline, so the amplitude is 5.
The maximum point is 0.5 units to the right of the minimum point, so the period is [tex]2\cdot 0.5 =1[/tex]. We multiply by 2 because the distance between consecutive minimum and maximum points is always [tex]\frac{1}{2}[/tex] of the period.
The general sinusoidal equation [tex]y=a\cos(bx)+d[/tex] is the result of horizontal and vertical shifts, reflections, and stretches of the parent equation [tex]y=\cos(x)[/tex] where
The amplitude is 5, so [tex]|a|=a=5[/tex].
The midline is y = 6, so d = 6.
The period is 1, so b = [tex]\frac{2\pi }{1}=2\pi[/tex].
The formula for [tex]g(x)=5\cos(2\pi x)+6[/tex].
