Graphing the Greatest Integer Function (3.13.2)

Question 1

• The greatest integer function looks for the greatest integer that is less than or equal to x in the notation [x].

Answer 1

• The greatest integer function looks for the greatest integer that is less than or equal to x in the notation [x].

Question 2

• The graph of this function is sometimes referred to as a “step-function” because of its appearance.

Answer 2

• The graph of this function is sometimes referred to as a “step-function” because of its appearance.

Question 3

The greatest integer brackets should be a tip off that some
sort of step function graph is appropriate.
If you graph the expression inside the brackets you can
visually see the general path that will be involved.
In this example, the expression graphs a line.
Notice the specific points that were located based on the
y-intercept and the slope of the expression. Each one is a
point where x has an integer value.
Now, think about all the values between the points shown.
The y for every point will be the largest integer that does not
exceed the f (x) at that point.
For example, when x = 2¼, 2x –1 = 3½ and [2x –1] = 3, the
largest integer not exceeding 3½. If you look on the graph,
you will see this point (2¼, 3) is graphed on the short blue
line that begins at (2, 2).
For this particular example, a new blue “step” will start
whenever x is an integer or attached to the fraction ½. That
is because every x-value multiplies with 2 and produces
integers.
Think about the point where x = 2½. At this point, 2x –1 =
4 and [2x –1] = 4. So, at the point (2½, 4) expect a new blue
line to start.
And, sure enough, one does.
The result is a graph composed of an ascending set of short
“steps,” each one covering half its integer space on the y-axis.

Answer 3

The greatest integer brackets should be a tip off that some
sort of step function graph is appropriate.
If you graph the expression inside the brackets you can
visually see the general path that will be involved.
In this example, the expression graphs a line.
Notice the specific points that were located based on the
y-intercept and the slope of the expression. Each one is a
point where x has an integer value.
Now, think about all the values between the points shown.
The y for every point will be the largest integer that does not
exceed the f (x) at that point.
For example, when x = 2¼, 2x –1 = 3½ and [2x –1] = 3, the
largest integer not exceeding 3½. If you look on the graph,
you will see this point (2¼, 3) is graphed on the short blue
line that begins at (2, 2).
For this particular example, a new blue “step” will start
whenever x is an integer or attached to the fraction ½. That
is because every x-value multiplies with 2 and produces
integers.
Think about the point where x = 2½. At this point, 2x –1 =
4 and [2x –1] = 4. So, at the point (2½, 4) expect a new blue
line to start.
And, sure enough, one does.
The result is a graph composed of an ascending set of short
“steps,” each one covering half its integer space on the y-axis.