Answer:
(d) 2sin(2θ +π/4) -1
Explanation:
The graphed function has a midline of -1 (eliminates choice A). It will be a cosine function shifted right π/8 units, or a sine function shifted left π/8 units.
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The shift is accomplished by replacing θ with (θ -π/8) in the sine function, or (θ +π/8) in the cosine function. That means, we're looking for an expression that is either of ...
2(cos(2(θ-π/8)) -1 = 2cos(2θ -π/4) -1 . . . . . no match
or
2(sin(2(θ+π/8)) -1 = 2sin(2θ +π/4) -1 . . . . . matches choice D
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Additional comment
In the form ...
g(x) = K·f(C(x -A)) +B
the factor K is a vertical stretch factor, C is a horizontal compression factor, A is a right shift, and B is an up shift. In this function, we have its amplitude stretched by a factor of 2, a down shift of 1 unit, and a horizontal shift of π/8. The horizontal compression factor is 2.
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The amount of horizontal shift for the cosine function is found at the peak nearest the y-axis. The amount of horizontal shift for the sine function is found at the upward midline crossing nearest the y-axis.