Questions: The coordinates of the five quarter points for y=cos x are given below. Use these quarter points to determine the y-coordinates of the quarter points of y=4 cos(4x). (0,1) (π/2, 0) (π/2, -1) (2π, 1) (Type ordered pairs, Type exact answers, using x as needed. Use integers or fractions for any numbers in the expression.)

Solution Steps

To determine the \( y \)-coordinates of the quarter points for \( y = 4 \cos(4x) \), we need to understand how transformations affect the cosine function. The function \( y = 4 \cos(4x) \) is a vertically stretched and horizontally compressed version of \( y = \cos x \). The amplitude is multiplied by 4, and the period is divided by 4. We will calculate the new quarter points by applying these transformations to the given quarter points of \( y = \cos x \).

The original quarter points for the function \( y = \cos x \) are given as:

• \( (0, 1) \)
• \( \left(\frac{\pi}{2}, 0\right) \)
• \( (\pi, -1) \)
• \( \left(\frac{3\pi}{2}, 0\right) \)
• \( (2\pi, 1) \)

The function \( y = 4 \cos(4x) \) involves a vertical stretch by a factor of 4 and a horizontal compression by a factor of \( \frac{1}{4} \). This means:

• The \( y \)-coordinates will be multiplied by 4.
• The \( x \)-coordinates will be divided by 4.

Applying the transformations to the original quarter points, we get:

• For \( (0, 1) \): \( \left(0, 4 \cdot 1\right) = (0, 4) \)
• For \( \left(\frac{\pi}{2}, 0\right) \): \( \left(\frac{\pi}{8}, 4 \cdot 0\right) = \left(0.3927, 0\right) \)
• For \( (\pi, -1) \): \( \left(\frac{\pi}{4}, 4 \cdot -1\right) = \left(0.7854, -4\right) \)
• For \( \left(\frac{3\pi}{2}, 0\right) \): \( \left(\frac{3\pi}{8}, 4 \cdot 0\right) = \left(1.1781, 0\right) \)
• For \( (2\pi, 1) \): \( \left(\frac{2\pi}{4}, 4 \cdot 1\right) = \left(1.5708, 4\right) \)

Final Answer