Answer:
[tex]y=8\cos \left(\dfrac{1}{2}x+\pi\right)+2[/tex]
Step-by-step explanation:
The equation of cosine function will be
[tex]y=a\cos (kx+b)+c[/tex]
Find [tex]a:[/tex]
The minimum y is [tex]y_{min}=-6,[/tex] the maximum y is [tex]y_{max}=10,[/tex] so
[tex]a=\dfrac{1}{2}\cdot |-6-10|=\dfrac{1}{2}\cdot 16=8[/tex]
Find [tex]k:[/tex]
The period of the function is [tex]T=4\pi,[/tex] so
[tex]T=\dfrac{2\pi }{k}\Rightarrow k=\dfrac{2\pi }{4\pi }=\dfrac{1}{2}[/tex]
Find [tex]b:[/tex]
At [tex]x=0,\ y=y_{min}=-6[/tex]
Function [tex]y=\cos x[/tex] at [tex]x=0[/tex] has the value [tex]y=y_{max}=1,[/tex] so [tex]b=\pi[/tex] [translate by one-fourth of the period]
Find [tex]c:[/tex]
The graph has the midline [tex]y=\dfrac{10-6}{2}=2,[/tex] so [tex]c=2.[/tex]
Hence,
[tex]y=8\cos \left(\dfrac{1}{2}x+\pi\right)+2[/tex]
