Describe the difference between the graph of y -3 = -tan(-2x + 180) and y = tan(x)

To describe the difference between the graph of the equation y - 3 = -tan(-2x + 180) and the graph of y = tan(x), we need to examine how each equation affects the behavior of the tangent function.

1. Vertical Shift: In the equation y - 3 = -tan(-2x + 180), there is a vertical shift upward by 3 units because the constant term -3 shifts the entire graph vertically. On the other hand, the equation y = tan(x) does not have any vertical shift.

2. Amplitude: The amplitude of the tangent function is not affected by either equation. The tangent function oscillates between negative and positive infinity, and this behavior remains unchanged in both equations.

3. Phase Shift and Reflection: In the equation y - 3 = -tan(-2x + 180), there is a phase shift and reflection due to the negative sign in front of the tangent function. The argument of the tangent function, (-2x + 180), indicates that the graph is shifted 180 degrees to the right. Additionally, the negative sign reflects the graph across the x-axis. Conversely, the equation y = tan(x) does not have any phase shift or reflection.

Therefore, the graph of y - 3 = -tan(-2x + 180) is vertically shifted upward by 3 units, phase-shifted 180 degrees to the right, and reflected across the x-axis compared to the graph of y = tan(x).