Answer:
[tex]y=3\cos(\frac{8}{7}x)+2[/tex]
Step-by-step explanation:
Recall
- General cosine function: [tex]y=a\cos(bx+c)+d[/tex]
- [tex]|a|[/tex] represents the amplitude, or the distance between the maximum/minimum of the function and the midline
- [tex]\frac{2\pi}{|b|}[/tex] represents the period, or the length of a cycle of the function which repeats continuously
- [tex]-\frac{c}{b}[/tex] is the phase shift/vertical shift
- [tex]d[/tex] is the midline, or the average between the maximum and the minimum, creating a horizontal center line
We are given that the amplitude is [tex]a=3[/tex] and that the equation of the midline is [tex]d=2[/tex]. Since we know the period to be [tex]\frac{7\pi}{4}[/tex], we need to solve for [tex]b[/tex] by setting up the equation [tex]\frac{7\pi}{4}=\frac{2\pi}{|b|}[/tex]:
[tex]\frac{7\pi}{4}=\frac{2\pi}{|b|}\\\\7\pi b=8\pi\\\\b=\frac{8}{7}[/tex]
Hence, our cosine function will be [tex]y=3\cos(\frac{8}{7}x)+2[/tex]. See the attached graph for a visual.
