In this module, you will explore step functions. We will be looking at two specific step functions, the floor (or rounding-down) function and the ceiling (or rounding-up) function. The three sliders change values in both functions. On the left hand side are the current values of each slider, as well as both functions, the floor function, fl(x), and the ceiling function, ce(x). The blue dot next to each item lets you toggle it on or off. The graph of each function is in the middle area, as well as the sliders. The sliders can be moved around by clicking and dragging to the left or right. On the right hand side is a table of domain (input) and range (output) for each function. You can scroll down to see more values. Play around with the sliders making sure to look at how the functions and table values change. When you are comfortable with the module, answer the questions below (you do not need to copy them down)!
1. What do you notice about the domain values in the table on the right as you move the sliders around? 2. What do you notice about the range values of the floor function in the table on the right as you move the sliders around? 3. What do you notice about the range values of the ceiling function in the table on the right as you move the sliders around? 4. Are there any domain values where the range for both the floor and the ceiling function are the same? If so, give a few examples. 5. The graph does not show open or closed circles. Draw a picture of one piece of the floor function, with the appropriate circles on each end. How did you know which circles to use? 6. Do the same as in 4., except with the ceiling function. 7. Move the sliders to , , and . Write down both functions and label where each slider variable is represented in each function. 8. How does sliding the slider change the graph/values of each function? 9. How does sliding the slider change the graph/values of each function? 10. How does sliding the slider change the graph/values of each function? 11. Is there any other type of line you can make by moving the sliders to specific positions? If so, describe what you did to create it? 12. Has this activity helped you make more sense of step functions? Be honest!