1. GREATEST INTEGER FUNCTION
When greatest integer acts on a number, the value that represents the result is the greatest integer that is less than or equal to the given number. There are several descriptors in that expression. First of all you are looking only for an integer. Secondly, that integer must be less than or equal to the given number and finally, of all of the integers that satisfy the first two criteria, you want the greatest one. The brackets which indicate that this operation is to be performed is as shown: ‘[ ]’.
Example: [1.97] = 1
There are many integers less than 1.97; {1, 0, -1, -2, -3, -4, …} Of all of them, ‘1’ is the greatest.
There are many integers less than -1.97; {-2, -3, -4, -5, -6, …} Of all of them, ‘-2’ is the greatest.
Example: [-1.97] = -2

2. Example: [6.31] = 6 Example: [-6.31] = -7 Example: [5] = 5
It may be helpful to visualize this function a little more clearly by using a number line.
-6.31
6.31 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7
Example: [6.31] = 6
Example: [-6.31] = -7
When you use this function, the answer is the integer on the immediate left on the number line. There is one exception. When the function acts on a number that is itself an integer. The answer is itself.
Example: [5] = 5
Example: [-5] = -5
Example:
Example:
If there is an operation inside the greatest integer brackets, it must be performed before applying the function.
Example: [5.5–3.6] = [1.9] = 1
Example: [ ] = [9.1] = 9
Example: [3.6–5.5] = [-1.9] = -2
Example: [5.53.6] = [19.8] = 19

3. Example: [5–2.99] = [2.01] = 2 Example: [5–3] = [2] = 2
1.99
Example: [5–2.99] = [2.01] = 2
Example: [5–3] = [2] = 2
Example: [5–3.01] = [1.99] = 1
The greatest integer function can be used to construct a Cartesian graph. The simplest of which is demonstrated below.
f(x) = [x]
To see what the graph looks like, it is necessary to determine some ordered pairs which can be determined with a table of values.
f(x) = [x] x
f(0) = [0] = 0
f(1) = [1] = 1 1
f(2) = [2] = 2 2
f(3) = [3] = 3 3
f(-1) = [-1] = -1 -1
f(-2) = [-2] = -2 -2
If we only choose integer values for x then we will not really see the function manifest itself. To do this we need to choose non-integer values.

4. f(x) = [x] x
f(0) = [0] = 0
f(0.5) = [0.5] = 0
0.5
f(0.7) = [0.7] = 0
0.7
f(0.8) = [0.8] = 0
0.8
f(0.9) = [0.9] = 0
0.9
f(1) = [1] = 1 1
f(1.5) = [1.5] = 1
1.5
f(1.6) = [1.6] = 1
1.6
f(1.7) = [1.7] = 1
1.7
f(1.8) = [1.8] = 1
1.8
f(1.9) = [1.9] = 1
1.9
f(2) = [2] = 2 2
f(-0.5) =[-0.5]=-1
-0.5
f(-0.9) =[-0.9]=-1
-0.9
f(-1) = [-1] = -1 -1

5. When all these points are strung together the graph looks something like this – a series of steps.
We can’t really state the last (most right) x-value on each step because there is always another to the right of the last one you may name. So instead we describe the first x-value that is NOT on a given step. Example: (1,0)
For this reason it is sometimes called the ‘STEP FUNCTION’.
Notice that the left of each step begins with a closed (inclusive) point but the right of each step ends with an open (excluding point)

6. Rather than place a long series of points on the graph, a line segment can be drawn for each step as shown to the right.
The graphs shown thus far have been magnified to make a point. However, these graphs are usually shown at a normal scale as you can see on the next slide.

7. f(x) = [x]
This is a rather tedious way to construct a graph and for this reason there is a more efficient way to construct it. Basically the greatest integer function can be presented with 4 parameters, as shown below.
f(x) = a[bx - h] + k
By observing the impact of these parameters, we can use them to predict the shape of the graph.

8. f(x) = [x]
a = 1
f(x) = 2[x]
a = 2
In these 3 examples, parameter ‘a’ is changed. As a increases, the distance between the steps increases.
f(x) = 3[x]
a = 3

9. f(x) = -[x]
a = -1
f(x) = -2[x]
a = -2
When ‘a’ is negative, notice that the slope of the steps is changed. Downstairs instead of upstairs. But as ‘a’ changes from –1 to –2, the distance between steps increases. The further that ‘a’ is from 0, the greater the separation between steps. This can be described with a formula.
Vertical distance between Steps = |a|

10. f(x) = [x]
b = 1
f(x) = [2x]
b = 2
As ‘b’ is increased from 1 to 2, each step gets shorter. Then as it is decreased to 0.5, the steps get longer.

11. = ab b > 0 b < 0 f(x) = [-x]
When ‘b’ is negative, notice that the slope of the steps and the orientation of each step changes. It now is open on the left but closed on the right – opposite to the way it is when ‘b’ is positive.
b > 0
b < 0
f(x) = -[-x]
a = -1
b = -1
Notice that when both ‘a’ and ‘b’ are negative the slope of the steps becomes positive again. Both parameters affect the slope.
Slope through closed points of each step
= ab
If ab > 0, steps are increasing.
If ab < 0, steps are decreasing.

12. Parameters h and k do not have an impact in defining the shape of the graph. These parameters simply translate the graph. This translation can be taken into account by determining a starting point. The previous four formulae can be used to construct the greatest integer function provided that there is an ordered pair to start with – the starting point. For this we can use the y-intercept, f(0).
Example:
2. Orientation of each step:
b > 0
1. Starting point: (0,5)
4. Vertical distance between Steps = |a|
=|2| = 2
5. Slope through closed
points of each step
= ab

13. b > 0 = ab 1. Starting point: (0,5) 2. Orientation of each step:
4. Vertical distance between Steps = |a|
=|2| = 2
5. Slope through closed
points of each step
= ab
After placing the first step, the closed point on the next step must be vertically aligned with the open point on the previous one, keeping in mind the slope of the steps (up or down).

14. b < 0 = |a| = ab f(0) = 3[-(0)-3]+5 =3(-3)+5 = -4
f(x) = 3[-x – 3] + 5
a = 3; b = -1; h = 3; k = 5
1. Starting point: (0,-4)
f(0) = 3[-(0)-3]+5
=3(-3)+5 = -4
2. Orientation of each step:
b < 0
4. Vertical distance
between Steps
=|3| = 3
= |a|
5. Slope through closed
points of each step
= ab

15. Five Steps to Construct Greatest Integer Graph
f(x) = a[bx - h] + k
1. Starting point: f(0)
4. Vertical distance
between Steps
= |a|
b > 0
b < 0
2. Orientation of each step:
Slope through closed points of each step
= ab