Answer:
[tex]2*cos(2x+\frac{\pi}{4} )-2[/tex]
Step-by-step explanation:
Recall the trigonometric definitions for the geometrical characteristics given to you:
For a General Harmonic function of the type: [tex]f(x)=A* cos(Bx+C)+D[/tex]
we define:
|A| = Amplitude of the function
Period of the function = [tex]\frac{2\pi }{B}[/tex]
Phase shift = [tex]-\frac{C}{B}[/tex]
vertical shift = D
Therefore we can construct a function that includes the appropriate geometric characteristics requested by using:
A = 2
To find B we use the definition of period, and what value we want it to have: [tex]\frac{2\pi }{B}=\pi \\\frac{2\pi }{\pi}=B\\B=2[/tex]
To find C we use the definition of phase shift and the value we want it to have (also using the value for B we found in the step above): [tex]-\frac{C}{B}=-\frac{\pi}{8} \\-\frac{C}{2}=-\frac{\pi}{8}\\C=\frac{2* \pi}{8} =\frac{\pi }{4}[/tex]
and finally, D = -2
Therefore the function will look like: [tex]2*cos(2x+\frac{\pi}{4} )-2[/tex]
