How do the graphs of y = 1/x and y =5/x+6 compare?
The graphs of y = 1/x and y = 5/x+6 are similar in shape, but they are horizontally shifted.
The graph of y = 1/x is a hyperbola that passes through the points (1, 1), (2, 0.5), (-1, -1), (-2, -0.5), etc. As x approaches positive or negative infinity, y approaches 0. It has two asymptotes: the x-axis (y = 0) and the y-axis (x = 0).
Meanwhile, the graph of y = 5/x+6 is a hyperbola that is horizontally shifted 6 units to the left compared to y = 1/x. It passes through the points (1, 11/7), (2, 16/14), (-1, 1/7), (-2, -4/14), etc. As x approaches positive or negative infinity, y approaches 6. This graph also has two asymptotes: the x-axis (y = 0) and a vertical line x = -6.
Therefore, the two graphs have the same shape, but one is shifted horizontally.