Y=1-3cosx

The equation \( y = 1 - 3\cos(x) \) describes a cosine function that has been vertically transformed. Here’s a breakdown of its characteristics:

1. Amplitude: The amplitude of the cosine function is determined by the coefficient of \(\cos(x)\). In this case, the amplitude is \(3\). This means the function will oscillate between two values.

2. Vertical Shift: The term \(1\) in the equation indicates a vertical shift of the graph. The base cosine function \(\cos(x)\) oscillates between \(-1\) and \(1\). By subtracting and adjusting with the coefficient, the graph of \(-3\cos(x)\) will oscillate between \(-3\) and \(3\). Adding \(1\) translates the entire graph up by \(1\), resulting in a new oscillation range of:

   \[ 1 - 3 = -2 \quad \text{(minimum value)} \]

   \[ 1 + 3 = 4 \quad \text{(maximum value)} \]

   So, the function \(y = 1 - 3\cos(x)\) oscillates between \(-2\) and \(4\).

3. Period: The period of the cosine function is \(2\pi\), which means it completes one full cycle every \(2\pi\) units along the x-axis.

4. Phase Shift: There’s no horizontal (phase) shift in this equation since there’s no additional constant added to \(x\) within the cosine function.

5. Graph Behavior: The graph of this function will start at its maximum value when \(x = 0\) (since \(\cos(0) = 1\)), reaching \(4\), then descending down to \(-2\) and returning to \(4\).

Summary:

• Function: \( y = 1 - 3\cos(x) \)
• Amplitude: \( 3 \)
• Vertical Shift: \( 1 \)
• Range: from \(-2\) to \(4\)
• Period: \(2\pi\)
• Phase Shift: None

If you need help with specific types of problems relating to this function, feel free to ask!