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Basic Transformations - Problem 2
University of Michigan
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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!
Basic graphic transformations so for this particular example we are going to look at the graph of –x². We already know what the graph of x² looks like it looks like a parabola so just a big 'u'. Okay so for this particular example we are going to plug in some points and see what happens when we throw that negative out in front.
So when we plug in -2 what you have to remember is your order of operations. You do your exponent before your negative so this actually has a little parenthesis right here so we want to do -2² turns into 4 and then the negative on the outside makes this -4. -1² is +1, negative on the outside turns this into -1, 0 -0 positive 0 doesn’t make a difference it also stays the same, 1² is 1 negative so we end up with -1 and -4. Okay so let’s plot some points and see what happens to this graph.
So we have (-2,-4) and again I’m not very concerned with position I just want to get the general idea of what’s happening (-1,-1), (0,0), (1,-1), (2,4). So we end up happening my graph is not through my points but you get the general idea of what the shape is like is we get a upside down parabola.
Compare this to what happened when we had our general x², graph just took a normal graph and flipped it over. So what the negative does on the outside is takes your graph flips it over the x axis this center line right here. Let’s take for example at one more of these transformations look at f(x) is equal to –x³.
We know that the x³ graph looks something like this. Again I’m not plotting my key points I’m not really concerned with the exact position of it, I just want to see what that negative does to it. What that negative does is that it flips everything over the x axis. So the stuff that used to be up here in the first quadrant is going to get flipped down and the stuff that’s below it, below the x axis gets flipped up. So what this graph is actually going to end up looking like is taking this and flipping it down this and flipping it up. It’s going to look like something like this. Your positive y values become negative, your negative y values become positive.
So whenever you are dealing with the transformation with a negative on the outside of your function just going to flip the graph over.
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