for a tangent graph is one of the asymptotes AWLAYS at - pi/2 ?
No, one of the asymptotes of the tangent graph is not always at -pi/2. The tangent graph has multiple asymptotes, and one of them occurs at -pi/2, but there are also other asymptotes.
To understand this conceptually, let's consider the graph of the tangent function. The tangent function has a repeating pattern where it oscillates between positive and negative infinity as it approaches certain angles. These angles occur when the input to the tangent function, measured in radians, is equal to (n + 1/2) * pi, where n is an integer.
For example, when n = 0, we get (0 + 1/2) * pi = pi/2, so the tangent graph has an asymptote at pi/2. However, when n = 1, we get (1 + 1/2) * pi = 3pi/2, so there is another asymptote at 3pi/2. This pattern continues for all n, resulting in multiple asymptotes at different angles.
Therefore, the tangent graph has asymptotes at pi/2, 3pi/2, 5pi/2, and so on. These asymptotes occur at intervals of pi/2 and are located in the second and fourth quadrants of the coordinate plane.
To summarize, the asymptotes of the tangent graph occur at angles of the form (n + 1/2) * pi, where n is an integer. While one of these asymptotes occurs at -pi/2, there are also other asymptotes spaced pi/2 units apart.