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Piecewise Functions - Problem 2
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!
Graphing a piecewise function that is divided into three regions: Really all we have to do n this case is figure out what our appropriate graph is in each domain that we’re looking at. We have one graph for x is less than -2, we have another graph for if x is between -2 and 1 and we have a third graph if x is greater than 1.
A couple of ways we can deal with this. If you have trouble visualizing where these breakdowns are you can draw sort of a dotted line, or “imaginary” line so you can see where one region stops and one region starts. You could also just keep in mind as your graphing it what regions you’re dealing with, which is what I’m going to do for this case.
We have our first line, f(x) is equal to 3, that’s just going to be a horizontal line and we use that if x is strictly less than -2. So x is -2 is right here, x is less than -2 is down from that. We know that we just have a horizontal line down from 3 up until 2, and I have an open circle at -2 because I am not equal 2. That takes care of our first graph.
We then go to our next one; 2x plus, 2x minus 1 if x is between -2 and 1. This is just a line with slope 2, y intercept -1. Two ways of doing with this one. What we could do is just graph this line all together. So we know we have a y intercept of -1 and we have a slope of 2, so this line looks something like that. But I’m only concerned with where x is greater or equal to -2 and less than 1. What I can do is go to my -2 point, that is going to be equal to, so this point is going to be on the graph and then erase everything that is below that point.
Similarly we go to our 1 value, x is 1 which is right here and that point is going to be on the graph. Sorry it will be on the graph but it won’t be on it because it will be an open circle and then we erase everything larger than 1. By graphing our graph as we know how to do and then just sort of chopping of the regions we’re not interested in we can then get this segment which is what we want.
The last graph we are concerned with is the –x² plus 1. This is in the region if x is greater than or equal to 1 so that’s going to be over in this region, but first I just need to plot this graph. It’s a parabola shifted upside down and moved up 1, so I know, let’s grab another color, that this graph is going to look something like that. Looking at the region we’re concerned with is just that x is greater than and equal to 1, if x is equal to 1, our y value is zero. So we have the point zero and then down. And then everything else, everything else where x is less than 1 is not going to be on this curve so we then have to get rid of it.
This gets filled in there and what we end up with is a piecewise graph in three different portions. Just graph each segment as you would normally and then just look at your domain to make sure you’re drawing the right graph in each portion of your window.
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