Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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Piecewise Functions - Problem 1

Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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Graphing a piecewise function. What I have behind me is a piecewise function. We have x² minus 2 for portion of the graph and x minus 2 for another portion of then graph. Really what we do is the domain restriction the restriction on when we use each graph is listed here so we are using x² minus 2 if x is less than or equal to 1. So what sometimes helps students to visualize this is to just draw sort of dotted vertical line that divides our graph into different regions.

I know that x equals 1 is my dividing line. What you can do and you don’t have to is to just draw a slight dotted invisible line so to see sort of way your barrier is going to be regarding when you go from one graph to the other. Now all we have to do is graph each graph as we normally would but just keep in mind that graph only occurs in its domain.

We use the graph of x² minus 2 when x is less than or equal to 1. That means when I am on the right of this dotted line, I use the graph x² minus 2. X² minus is just a parabola shifted down 2 units, so I know that down 2 units, that’s one, let's go to two. We know that we have a parabola that comes down and I often times plug in my end point just to see where the graph is going to end up. If x is 1 we end up with 1², 1 minus 2, -1. I know that my graph is going to end right here. I can put a solid hole because my graph is going to be equal to at that point. When x is 1, I use this curve. That’s going to come up to this point. I’ve taken care of that first part of my piecewise graph. When x is less than or equal to 1 we use or parabola.

The next thing we want to go is if x is greater than 1. That’s referring to the right of this line. X minus 2 is just a line shifted down 2, so we know that line is just going to be something like this slope of 1. To find out exactly where it is when x is equal to 1, we can plug 1 in, 1 minus 2 is -1. Again we actually end up at -1. If we were to graph this line properly we would actually end up with an open circle because we do not include this point. It already is on the graph so it really doesn’t matter but I’m just going to throw it in there just so you can see the difference and then we have a slope of 1. So something like that.

Dealing with your domain restrictions to figure out which graph you use is which region. Look for when it’s less than that tell you you’re going to be graphing on the left side, greater than you’re going to be graphing on the right and then just graph the appropriate graph in the proper region.