You have been told and shown many times that mathematics is meant to display and analyze the world around us. We can use functions to give guesses for future happenings of a situation or we can use it to see patterns that have been happening, but what happens one one function alone cannot give the correct picture of what is going on. What if the behavior of a situation changes after time? Can math still help with situation or are we out of luck?

Have no fear! Math can still help! [br][br]When we have situations that act in different ways in different times, we use what mathematicians call a piece-wise function. [br][br]Consider when we buy items in bulk. Often an owner will make the price cheaper as you get into bigger and bigger amounts of their product. For example, let's say that I am a baker. I will charge 10 dollar for a pound of my cookies until you buy more than a 5 pounds, then every pounds of cookies more than that will be 5 dollars. Let's take a look at what is going on in the situation. Below shows the two prices as being a slope of a line. Notice the the purple line with a slope of 5 has half of the slope of the other red line which has a slope of 10.

This picture shows us how the function does change depending on the amount of pounds of cookies that you are buying, but notice it doesn't align with the story and how it explains when the prices change and how it still keeps the previous price into account when deciding their total bill. On the graph below try to determine how we can move the purple line so that it works with my pricing scheme better. The graph below shows the two lines and it will tell you the distance point a is from the x-axis. You will try to move point A so that the line that goes through A also goes through point D

You should have seen that the correct distance from the x-axis should be 25. We can remember that that distance is called the y-intercept and it is the b in the mx+b form of a line. The two lines that we think best describe the situation are shown below in the graph below along with their respective equations.

Now that we have figured out the functions that best describe our situation we need to figure out how to effectively write it as a piece-wise function. To do this, we need to decide what x-values need to be using which function. Play with the graph below, and determine how we should write our piece-wise function.