How do the graphs of y=1/x and y=5/x+6 compare?
the second is the first, expanded by 5 and shifted up by 6.
To compare the graphs of y=1/x and y=5/x+6, we can start by analyzing their equations.
The equation y=1/x represents a hyperbola. It will have an asymptote at the x-axis (y=0) and the y-axis (x=0). The graph will approach these asymptotes but will never intersect them.
On the other hand, the equation y=5/x+6 is a translated version of y=1/x. The "+6" term shifts the graph 6 units upwards, so the asymptote at the y-axis will be at y=6. This translation moves the entire graph up.
Comparing the two graphs, we see that the graph of y=5/x+6 is higher than y=1/x because of the vertical translation. Furthermore, the graph of y=5/x+6 still has asymptotes at the x-axis and y=0, just like y=1/x.
To visualize the comparison, we can plot both graphs on a graphing calculator or software. By plotting the points, we can clearly see how the two graphs differ in terms of height and position.