Determine how each function is related to graph of f(x) = |x|.  f(x) = 2|x|  f(x) = |x-1|  f(x) = |x| + 3

Greatest Integer Function – the most notable step function Step Functions – functions whose graphs resemble sets of stair steps. Notation of Greatest Integer Function : x Meaning of Greatest Integer Function : the greatest integer less than or equal to x.

It may be helpful to visualize this function a little more clearly by using a number line. When you use this function, the answer is the integer on the immediate left on the number line Exception: When you evaluate an exact integer, like 3, the answer is the integer itself.

Let’s evaluate some greatest integers… 2 = 9 = -3 = 2 ? 9 ? ? the greatest integer less than or equal to x.

Let’s evaluate some greatest integers… 2.2 = 1/21/2 = -4.1 = 2? 0 ? -5 ? the greatest integer less than or equal to x

Let’s evaluate some greatest integers… 9.1 = 51/351/3 = -2 2 / 9 = 9 ? 5 ? -3 ? the greatest integer less than or equal to x

If there is an operation inside the greatest integer brackets, it must be performed before applying the function.

xy Now that you know how to evaluate greatest integer functions, you can graph them. y = x +2

xy Now that you know how to evaluate greatest integer functions, you can graph them. y = x - 4

xy Now that you know how to evaluate greatest integer functions, you can graph them. y = -x + 2

When all these points are strung together the graph looks something like a series of steps. Reasoning for: ‘STEP FUNCTION’. Notice that the left of each step begins with a closed point but the right of each step ends with an open point We can’t really state the last 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)

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. f(x) = [x]

Wheels Bike Rentals charges a $6.00 flat rate and $1.50 for each hour you rent a bicycle including fractions of an hour (For example, 3.5 hours is $1.50(3) + $6.00). Use the greatest integer function to create a model for the cost C of renting a bicycle for x hours. Sketch the graph for up to 5 hours. xC