44. \( y=-4 \cos \left(x+\frac{\pi}{4}\right) \)

The equation \( y = -4 \cos \left(x + \frac{\pi}{4}\right) \) represents a cosine function that has been transformed in several ways. Let's analyze the transformations step by step: 1. **Amplitude**: The amplitude of the cosine function is given by the coefficient in front of the cosine. Here, the amplitude is \( | -4 | = 4 \). This means the graph will oscillate between \( 4 \) and \( -4 \). 2. **Period**: The standard period of the cosine function \( \cos(x) \) is \( 2\pi \). Since there is no horizontal stretching or compression factor (the coefficient of \( x \) inside the cosine is \( 1 \)), the period remains \( 2\pi \). 3. **Phase Shift**: The term \( \frac{\pi}{4} \) inside the cosine function indicates a phase shift. The phase shift can be calculated as: \[ \text{Phase Shift} = -\frac{\pi}{4} \] This means the graph is shifted to the left by \( \frac{\pi}{4} \). 4. **Vertical Shift**: There is no vertical shift in this function since there is no constant added or subtracted outside the cosine function. 5. **Reflection**: The negative sign in front of the amplitude indicates a reflection across the x-axis. This means that the peaks of the cosine function will become troughs and vice versa. ### Summary of Transformations: - Amplitude: \( 4 \) - Period: \( 2\pi \) - Phase Shift: Left by \( \frac{\pi}{4} \) - Reflection: Across the x-axis ### Graphing the Function: To graph \( y = -4 \cos \left(x + \frac{\pi}{4}\right) \): - Start with the basic cosine function \( y = \cos(x) \). - Reflect it across the x-axis. - Shift it left by \( \frac{\pi}{4} \). - Scale the graph vertically by a factor of \( 4 \). ### Key Points: - The maximum value of the function is \( 0 \) (at \( x = -\frac{\pi}{4} + 2k\pi \), where \( k \) is an integer). - The minimum value of the function is \( -4 \) (at \( x = -\frac{\pi}{4} + (2k+1)\pi \)). - The function will cross the x-axis at points where \( \cos \left(x + \frac{\pi}{4}\right) = 0 \). This analysis provides a comprehensive understanding of the function \( y = -4 \cos \left(x + \frac{\pi}{4}\right) \).