Answer:
[tex]f(x)=-2\cos(2x)+1[/tex]
Step-by-step explanation:
Recall the general cosine equation
- Function: [tex]f(x)=a\cos(bx+c)+d[/tex]
- Amplitude: [tex]|a|[/tex]
- Period: [tex]\frac{2\pi}{|b|}[/tex]
- Vertical Shift: [tex]-\frac{c}{b}[/tex]
- Midline: [tex]y=d[/tex]
Identify amplitude
[tex]\text{Amplitude}=\frac{\text{Max-Min}}{2}=\frac{3-(-1)}{2}=\frac{4}{2}=2[/tex]
Identify period and solve for b
[tex]\frac{3\pi}{2}-\frac{\pi}{2}=\pi\\ \\\frac{2\pi}{|b|}=\pi\\ \\2\pi=b\pi\\\\2=b[/tex]
Identify midline
[tex]y=d=1[/tex]
Final Equation
[tex]f(x)=-2\cos(2x)+1[/tex]
Also, the reason why [tex]a=-2[/tex] is because a cosine function starts at its maximum, but since it starts at its minimum, the value of [tex]a[/tex] must be negative and causes the wave to flip about the midline.
