Final answer:
The functions y=x and y=x-3 have the same slope but different y-intercepts. The graph of y=x-3 is shifted downward by 3 units compared to the graph of y=x.
Step-by-step explanation:
When comparing the functions y=x and y=x-3, we can see that they have the same slope, which is 1. This means that for every increase of 1 unit in the x-value, the y-value also increases by 1 unit. However, the y-intercepts of the two functions are different. The function y=x has a y-intercept of 0, while the function y=x-3 has a y-intercept of -3. This means that the graph of y=x-3 is shifted downward by 3 units compared to the graph of y=x.
These two equations are closely related in that they have the same slope, which determines the steepness of the line, but they have different y-intercepts. The y-intercept is where the line crosses the y-axis. The function y = x crosses at the origin (0,0) while y = x - 3 crosses the y-axis at the point (0,-3). The second function is essentially the first function shifted downward by 3 units.
To observe this relationship graphically, one could plot both functions on a graph. By choosing different values for x, calculate the corresponding y values for both functions and plot these points. The resulting graphs would show that both lines are parallel, indicating that they have the same slope and will never intersect.