Question

Consider y=h(θ) described in the table. When comparing h(θ) to the parent function f(θ)=cos(θ), which statements are true? Select all that apply.

Answer 1

2 Answers:

• Choice A
• Choice E

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Explanation:

Let's go through the answer choices

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A)

The lowest point is y = 1, and the highest point is y = 27. Computing the midpoint gets us (yMin+yMax)/2 = (1+27)/2 = 28/2 = 14.

The midline of $h(\theta)$ is y = 14.

Consider that the parent function $\cos(\theta)$ has a midline of y = 0. The jump from y = 0 to y = 14 must mean $h(\theta)$ has a vertical shift of 14.

Choice A is true

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B)

The period is the length of the cycle. For the function $h(\theta)$, the function starts at $\theta = 0$ and repeats when it gets to $\theta = \frac{\pi}{2}$. This is a difference of pi/2 units which is the period.

Choice B is false. It contradicts with choice E.

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C)

Choice C is false. The max for $h(\theta)$ is at y = 27

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D)

To figure out the amplitude, we can note the vertical distance from the midline y = 14 to the max is 27-14 = 13 units.

Or we can note the distance from the midline to the min is 14-1 = 13 units.

Or we subtract the min and max, and divide by 2:  (yMax-yMin)/2 = (27-1)/2 = 26/2 = 13

Either way, the amplitude is 13 units

Choice D is false.

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E)

Choice E is true. Refer to choice B to see why this is the case.

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Side note: one possible function is $h(\theta) = 13\cos(4\theta+\pi) + 14$. The various pieces of that function (in)directly lead to the amplitude, period, and vertical shift.