Question

Which of the following functions is represented in the graph shown?

f(x) = −2cos(2x) + 1
f(x) = −4cos(2x) − 1
f(x) = −2cos(x) − 1
f(x) = −4cos(x) + 1

Answer 1

`f(x)=-2\cos(2x)+1`

Explanation:

Recall the general cosine equation

• Function:
  `f(x)=a\cos(bx+c)+d`
• Amplitude:
  `|a|`
• Period:
  `(2\pi)/(|b|)`
• Vertical Shift:
  `-(c)/(b)`
• Midline:
  `y=d`

Identify amplitude

`\text{Amplitude}=\frac{\text{Max-Min}}{2}=(3-(-1))/(2)=(4)/(2)=2`

Identify period and solve for b

`(3\pi)/(2)-(\pi)/(2)=\pi\\ \\(2\pi)/(|b|)=\pi\\ \\2\pi=b\pi\\\\2=b`

Identify midline

`y=d=1`

Final Equation

`f(x)=-2\cos(2x)+1`

Also, the reason why
`a=-2` is because a cosine function starts at its maximum, but since it starts at its minimum, the value of
`a` must be negative and causes the wave to flip about the midline.