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Graphing the Transformation y = f(x - h) - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Transformations of the function y equals 1 over x are a little trickier, because this function has asymptotes. I have an example here. Y equals -2 over x plus 3 plus 1. Now note that, this is the same as -2 times 1 over x plus 3 plus 1. So it's clear that 1 over x is the function that's being transformed. So I'll write that down; f(x) equals 1 over x.
Recall what the graph of that function looks like. You've got a hyperbola, two branches, and it's got these asymptotes. These are lines that the graph gets closer and closer to as you move away to the right, to the left, up or down. So we'll have to pay attention to what happens to these asymptotes as we graph. This asymptote is s equals 0, this one is y equals 0. In fact I'll write that down on my table. X equals 0, and y equals 0.
Let's make a table of values for our basic parent function u, 1 over u. So let's plug in easy points like -2, -1, 1, and 2. The reciprocal of -2 is -½, -1, 1, and ½. Now for this function I'm going to make the substitution u equals x plus 3. U equals x plus 3 means x equals u minus 3.
So I'm going to take these u values, and subtract 3 from them to get my x values. So I subtract 3 from -2, and I get -5. Subtract 3, -4. Subtract 3 from 1 I get -2. Subtract 3 from 2, I get -1. So my x values are all moving three units to the left.
Now what happens to the y values? Well the y values remember x plus 3 is u, so 1 over u has to be multiplied by -2, and then I have to add 1. So I'm going to multiply each of these by -2, and add 1. - ½ times -2 is 1, plus 1 is 2. -1 times -2 is 2 plus 1 is 3. 1 times -2 is -2, plus 1 is -1. ½ times -2 is -1 plus 1 is 0.
One more thing, let's find out what happens to these asymptotes. Now x equals 0, this is an x value, and so whatever I did to these x values to transform them, I can also do with this asymptote. I subtracted 3, x equals u minus 3, so I can also subtract 3 from this guys. I get x equals -3. That's my vertical asymptote.
The y values I multiplied by -2, and added 1. So I can do the same thing here. Multiply this y value by -2, I get 0 plus 1 is 1. So you can apply your transformations to the asymptotes as well. Just remember which is which.
This is an x value, so apply the horizontal transformation there. A y value, you apply the vertical transformation there. Let's plot these asymptotes right away. So we've got x equals -3, and y equals 1. Well -3 is right here, so here is x equals -3. Y equals 1 is right here.
Now let's plot our points. So we've got -5, 2 that's -3 so -4, -5, 1, 2 that' s a point. -4, 3. -4 1, 2, 3, and we're going to have some thing like this. It's still going to have the same basic shape as the original y equals 1 over x graph. This is just a transformation of this graph. -2, 1. -2 is here, -1 is here. -1, 0 is right here. So we're going to have something like this. That's it. That's your transformed reciprocal function graph.
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