Question

Write the equation of the trigonometric graph.

Answer 1

`y=\boxed{2}\:\cos \left(\boxed{1}\;x\right)+\boxed{3}`

Explanation:

The graph of the solid black line is the cosine parent function, y = cos(x).

The standard form of a cosine function is:

`\boxed{y = A \cos(B(x + C)) + D}`

where:

• A is the amplitude (height from the mid-line to the peak).
• 2π/B is the period (horizontal distance between consecutive peaks).
• C is the phase shift (horizontal shift - positive is to the left).
• D is the vertical shift (the mid-line is y = D).

From inspection of the graph, the x-values of the turning points (peaks and troughs) of the parent function and the new function are the same. Therefore, the period of both functions is the same, and there has been no horizontal shift. So, B = 1 and C = 0.

The mid-line of the new function is y = 3. Therefore, D = 3.

The y-value of the peaks is y = 5. The amplitude is the distance from the mid-line to the peak. Therefore, A = 2.

Substituting these values into the standard formula we get:

`y = 2 \cos(1(x + 0)) + 3`

`y=2 \cos (1(x))+3`

`y= 2 \cos(x) + 3`

Therefore, the equation of the trigonometric graph is:

`y=\boxed{2}\:\cos \left(\boxed{1}\;x\right)+\boxed{3}`