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.
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?
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?