Y=3 cos x

The equation \( y = 3 \cos x \) describes a cosine wave that has been vertically stretched and does not include any phase shifts or horizontal stretches/compressions. Here are some details about the function: 1. **Amplitude**: The amplitude of the cosine function is the coefficient in front of the cosine. In this case, the amplitude is \( 3 \). This means the graph of the function will oscillate between \( 3 \) and \( -3 \). 2. **Period**: The period of the standard cosine function \( \cos x \) is \( 2\pi \). Since there is no coefficient affecting \( x \) (which would stretch or compress the wave horizontally), the period remains \( 2\pi \). 3. **Phase Shift**: There is no horizontal shift since there is no addition or subtraction inside the cosine argument (i.e., there is no \( \cos(kx + c) \) form). 4. **Vertical Shift**: There is no vertical shift since there is no constant added or subtracted from the cosine function. ### Key Points of the Graph: - The maximum value of \( y \) is \( 3 \) (occurs when \( \cos x = 1 \)). - The minimum value of \( y \) is \( -3 \) (occurs when \( \cos x = -1 \)). - The function crosses the x-axis (where \( y = 0 \)) at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. ### Plotting the Graph: To plot the graph, you can follow these steps: 1. Draw the x-axis and y-axis. 2. Mark the amplitude limits at \( +3 \) and \( -3 \). 3. Start at \( (0, 3) \), since \( \cos(0) = 1 \). 4. Mark points for one full period: \( (0, 3) \), \( \left( \frac{\pi}{2}, 0 \right) \), \( (\pi, -3) \), \( \left( \frac{3\pi}{2}, 0 \right) \), and \( (2\pi, 3) \). 5. Draw a smooth, continuous wave that peaks at \( 3 \) and dips to \( -3 \). This gives a complete picture of the function \( y = 3 \cos x \). If you need further explorations such as derivatives or integrals associated with this function, let me know!