Enter your answer and show all the steps that you use to solve this problem in the space provided.
How do the graphs of y=1/x and y=5/x+6 compare?
To compare the graphs of y=1/x and y=5/x+6, we can analyze their key characteristics and then plot them on a graph to visualize the differences.
First, let's look at the general shapes of the two functions. The function y=1/x represents a hyperbola that touches the x and y axes at their respective points. The function y=5/x+6 is also a hyperbola but is vertically shifted up by 6 units and vertically stretched by a factor of 5.
Next, let's analyze the asymptotes of the two functions. The function y=1/x has vertical and horizontal asymptotes. The vertical asymptote occurs at x=0, and the horizontal asymptote is the x-axis (y=0). For y=5/x+6, the vertical asymptote is still at x=0 but the horizontal asymptote has shifted up by 6 units and is now the line y=6.
Now, let's plot the graphs of y=1/x and y=5/x+6 on the same set of axes. To do this, it's helpful to choose some x-values and calculate the corresponding y-values for each function. For example, for y=1/x, let's choose x=1, 2, and 3. The corresponding y-values are y=1, 0.5, and 0.33, respectively. For y=5/x+6, when x=1, the calculated y-value is 11, and so on.
Plotting these points and connecting them with smooth curves will give a better visualization of how the two functions compare. The graph of y=1/x will show a hyperbola that touches the x and y axes at their respective points, while the graph of y=5/x+6 will be a vertically stretched and shifted hyperbola with a different asymptotic behavior.
In conclusion, the key differences between the graphs of y=1/x and y=5/x+6 lie in their transformations, asymptotic behavior, and shape. The graph of y=5/x+6 will be a vertically stretched and shifted hyperbola compared to the graph of y=1/x.