Question

Solve the following problem and choose the best answer.
Question
W
The xy-plane above shows a portion of the graph of which of the following functions?
○ y =
3 cos 2x
y = 3 cos 2x
y=-3 cos (x-π)
y 3 cos (x-π) -

Answer 1

Answer:

B)  y = 3 cos 2x

Step-by-step explanation:

The standard form of a cosine function is:

$y= A \cos(B(x + C)) + D$

where:

• A represents the amplitude.
• B represents the frequency, and 2π/|B| is the period.
• C represents the phase shift (horizontal translation).
• D represents the vertical shift (vertical translation).

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Amplitude

The amplitude of a cosine graph is half the positive difference between the function's maximum and minimum values.

In this case, the maximum value of the graphed function is y = 3, and the minimum value of the function is y = -3. Therefore:

$A=\dfrac{|3-(-3)|}{2}=3$

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Frequency

The period of a cosine graph is the horizontal distance between consecutive peaks (or troughs).

From observation of the given graph, the period of the function is π. This means that:

$\dfrac{2\pi}{|B|}=\pi \implies B=2$

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Horizontal Shift

The parent cosine function y = cos⁡(x) is an even function, indicating symmetry about the y-axis. Its maximum value occurs at x = 0, corresponding to its y-intercept. Since the graphed function retains this symmetry about the y-axis and its maximum is also at x = 0, there is no horizontal shift evident.

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Vertical Shift

The midline of a cosine graph is given by the equation y = D, where D represents the y-value of the midpoint between the maximum and minimum values of the function. Therefore:

$D=\dfrac{3+(-3)}{2}=0$

So, the midline of the graphed function is y = 0. This indicates that there has been no vertical shift.

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Equation of the function

To write the equation the function, substitute A = 3, B = 2, C = 0 and D = 0 into the general formula:

$y= 3 \cos(2(x + 0)) + 0$

This simplifies to:

$\Large\boxed{\boxed{y= 3 \cos(2x)}}$