Answer:
y = 2cos(x) +1.5cos(2x) +2.5
Step-by-step explanation:
You want the equation that produces the graph shown.
Symmetry
The function is symmetrical about the y-axis, and it has a period of 2π. The bumps at multiples of x=π suggest there is a component of the function that has a period of π.
Cosine
The cosine trig function is an even function (symmetrical about the y-axis) and periodic with a period of 2π. The function y=cos(2x) will be a horizontal compression by a factor of 2, so will have a period of π.
The cosine function has an average value of 0, so the function in this graph has a vertical translation of some amount.
Equation
We can assume the equation will be of the form ...
y = a·cos(x) +b·cos(2x) +c . . . . . . . where c provides the vertical translation
It is convenient to use the peak values to find the coefficients. In addition, we can use the function value at x = π/2 to help find the vertical translation.
Evaluating the function at x=0 gives ...
6 = a·(1) +b·(1) +c
Evaluating the function at x=π gives ...
2 = a·(-1) +b·(1) +c
Evaluating the function at x=π/2, we have ...
1 = a·(0) +b·(-1) +c
Solution
Now, we have 3 equations in 3 unknowns that we can solve for the coefficients.
Subtracting the second equation from the first gives ...
(6) -(2) = a(1 -(-1)) +b(1 -1) +c(1 -1)
4 = 2a ⇒ a = 2
Adding the second and third equations, we have ...
(2) +(1) = a(-1 +0) +b(1 -1) +c(1 +1)
3 = -2 +2c ⇒ (3 +2)/2 = c = 5/2
Substituting this into the third equation gives ...
1 = -b +5/2 ⇒ b = 5/2 -1 = 3/2
Using these coefficients, the equation becomes ...
y = 2cos(x) +1.5cos(2x) +2.5
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Additional comment
The attached graph shows this matches the given graph pretty well.
The attachment shows the cosine function and its horizontally compressed version for your reference. Since the given graph uses angles in radians, that is the angle measure we used above. The translation to degrees is cos(π/2) = cos(90°) = 0, cos(π) = cos(180°) = -1.
