Question

Match the amplitude, midline, period, and frequency for the cosine equation

Answer 1

Step-by-step explanation

we can describe the cosine function as

`y=A\cos \mleft(B\mleft(x+C\mright)\mright)+D`

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

`5\cos (2x)+3\rightarrow A\cos (B(x+C))+D`

hence

`\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}`

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

`\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}`

b) we have a minimum when

`\begin{gathered} \cos (2x)=-1 \\ x=(\pi)/(2),\text{ because} \\ \cos (2(\pi)/(2))=\cos (\pi)=-1 \\ \text{now, replace} \\ y=5\cos (2\cdot(\pi)/(2))+3=5\cdot-1+3=-5+3=-2 \end{gathered}`

so, the midline is the average of 8 and -2

`\begin{gathered} \text{midline}=y=(8+(-2))/(2)=(6)/(2)=3 \\ y=3 \end{gathered}`

I hope this helps you