Library+of+Functions-Target+D-Stepping+it+Up-Guided+Learning

**Target D: Graph a Step-Function within Contextual Problem**
We are going to solve contextual problems whose graphs are step functions. What is a step function? A step function is a piecewise constant function whose graph is a series of segments. The graph looks like "steps". Let's look at a graph of a step function.

If you take a look at the graph, it does actually look like a bunch of steps. Each step is an individual constant function. You will notice a few things about this graph. It makes jumps each time it gets to the next integer x value. If we evaluate the function at f(1) you should notice there is an open circle at y = 0 and a closed circle at y = 1. From our past experiences with graphing, f(1) = 1 because the closed circle means the graph has a value at that point whereas the open circle means that y value is not included at that particular point. This particular function is called the greatest integer function.

Now let's do a problem in context.

You are going to rent a Segway to tour around Chicago. It is a $10 flat fee to rent the vehicle, as well as, $12 per each hour or partial hour, each fraction of an hour will be rounded up to the next hour. Graph the step-function if you have a total of $75 to spend. How much will it cost if you rent for 3.5 hours? Can your rent for 6 hours? Let's look at the table and graph below.

It will cost you $58 for 3.5 hours. And you could not rent for 6 hours because it would cost $82 and you only have $75.

Quick Check
You are on vacation in South Haven and you want to rent a paddle board to get a little exercise while enjoying the lake. You have a total of $150 to spend on this adventure. It costs $30 per hour to rent the paddle board, each partial hour will be rounded up to the next hour. They also charge a non-refundable fee of $15 for renting the paddle board. Create a graph to represent the situation. How much will it cost for 2.5 hours? Can you rent the paddle board for 7 hours?