Step Function

1. Greatest Integer Function

2. GIF: Step Function
 The Greatest Integer Function (GIF) or
Step Function is used in businesses as a
way of rounding numerical values down.
 It is symbolized by square brackets [ x ],
but on your graphing calculator by the
math function int(x).
 E.g. [1.3] = 1
 E.g. [2.9] = 2
 E.g. [-3.4] = -4 (because it rounds DOWN)

3. Graph of Function
 The graph appears as a set of stairs,
hence step function.
 You will notice that the beginning of each
step is a clear or full circle.
 The clear circle indicates that the value of
x is NOT included.
 The full circle indicates that the value of x
IS included.

4. Graph

5.  Properties of Basic and Transformed GIF
Rule y = [x] y = a [b (h - k)] +k
Graph Series of ascending
steps
Depends on a,b
Domain R R
Range ]-∞,∞[ Depends
Zero None Depends
Extremes None None
Sign Positive {x≥0} Depends on a,b,h,k
Variation Increasing Depends

6. Parameters
 ║a║  the height of the step
 b  1/b give the length of the step
 h  horizontal translation of starting point
 k vertical translation of starting point
 The starting point is hypothetical, however
in most everyday situations, we only deal
with positive numbers on the x and y axes.

7. Standard Format
 The standard format is y = a[b(x-h)]+k
 You must change the given formula into
this format.
 You cannot multiply a with b and remove it
from the brackets.
 ‘a’ gives the height of the steps.
 ‘b’ give the 1/b which gives the length of
the step.

8. Example
 Given: y = 3 [0.5x]:
 We know the height of the step is 3
 We know 1/b = 1/0.5 = 2 is the length of
the step.
 ‘a’ is positive so the first value is included
and the last value is excluded.

9. Graph y = 3 [0.5x]
y
8
7
6
5
4
3
2
1
0 1 2 3 4 5 6 7 x

10. Activity
 P. 30
 Q. 1,2,3,6,8