Explanation
we can describe the cosine function as
[tex]y=A\cos \mleft(B\mleft(x+C\mright)\mright)+D[/tex]
where
amplitude is A
Frequency is B
period is 2π/B
phase shift is C (positive is to the left)
vertical shift is D
Step 1
identify
[tex]5\cos (2x)+3\rightarrow A\cos (B(x+C))+D[/tex]
hence
[tex]\begin{gathered} A=5=\text{Amplitude} \\ B=2,C=0,so \\ \text{period}=\frac{2\text{ }\pi}{B}=\frac{2\text{ }\pi}{2}=\pi \\ \text{period}=\pi \\ Frequency=B=2 \\ \text{Vertical shift=D=3} \end{gathered}[/tex]
Step 2
midline
The equation of the midline of periodic function is the average of the maximum and minimum values of the function.
a) we have a maximum when
[tex]\begin{gathered} \cos (2x)=1 \\ x=0,\text{ because (cos 0)=1} \\ \text{now, replace} \\ y=5\cos (2x)+3 \\ y=5\cos (2\cdot0)+3=5\cdot1+3=8 \\ y=8,\text{ so the max. is 8} \end{gathered}[/tex]
b) we have a minimum when
[tex]\begin{gathered} \cos (2x)=-1 \\ x=\frac{\pi}{2},\text{ because} \\ \cos (2\frac{\pi}{2})=\cos (\pi)=-1 \\ \text{now, replace} \\ y=5\cos (2\cdot\frac{\pi}{2})+3=5\cdot-1+3=-5+3=-2 \end{gathered}[/tex]
so, the midline is the average of 8 and -2
[tex]\begin{gathered} \text{midline}=y=\frac{8+(-2)}{2}=\frac{6}{2}=3 \\ y=3 \end{gathered}[/tex]
I hope this helps you
