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!