Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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Integer Power Functions - Problem 2

Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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I want to graph another transformed power function; this time y equals -1/8 times the quantity x minus 2 to the 5th power plus. We have a lot of transformations happening here.

First let’s identify that the basic power function at work here is y equals x to the 5th so I need a table of values for, and I’ll change the variable to u so that I can transform in a second.

The values that we need here, let’s choose easy ones to start with; -1, 0 and 1. When you raise -1, 0 or 1 to eh 5th power you actually get the same thing in each case, -1, 0, 1. Now here I want to make a substitution, I want to come up with values for my transformed function and I want to make a substitution for x minus 2. U equals x minus 2. So x equals u plus 2.

This substitution allows me to see what I need to do to these u values in order to get my x values. What kind of transformation’s happening there? I’m adding 2 so I’m shifting to the right. Add 2, add 2, add 2, and then for the y values, I multiply the u to the 5th times -1/8 and then I add 1. Times -1/8 is 1/8, plus 1,t hat’s 9/8. Times -1/8 is zero plus 1 is 1. Times -1/8 is -1/8 plus 1 is 7/8.

These points look like they’re not quite going to be enough to give me a good graph so I can extend this table backwards and plot more points. So -2 to the 5th power is -32 and 2 to the 5th is 32. Remember the x coordinates I need to add 2 so -2 plus 2 is zero, 2 plus 2 is 4 and then the y coordinates I multiply by -1/8 and add 1. -32 times -1/8 is 4 plus 1 is 5. 32 times -1/8 is -4 plus 1 is -3. Let’s plot these points. Start with (2, 0) it’s important to know, remember the shape of y equals x to the 5th. It looks a lot like x to the 3rd, something like this and so the (0, 0) point is really important. It’s the inflection point of the graph where the graph has a twist. This point has moved from (0, 0) to (2, 1).

It’s important to know that that’s your new inflection point, (2, 1). There should be a twist here. And then let’s plot the other points, (0, 5) is up here, (4, -3) is down here. Just to get some shape let’s use the other points. (1, 9/8) that’s 1 and 1/8, just a little above 1 and then (3, 7/8), that’s 3 and then you have to go 1/8 lower than 1. Something like that. So our shape is going to be like this, really steep down, and again.

That’s it. Remember there’s a little twist here that happens at the inflection point. This is our graph of y equals -1/8, quantity x plus 2 to the 5th plus, x minus 2 to the 5th plus 1.

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