PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to graph another transformed power function. Y equals 10 times the quantity 2/5 x minus 1 to the 3rd power plus 10. There’s a lot of transformations but what’s the basic function that’s being transformed? It’s y equals x to the 3rd. I want a table of u and u³ and those are the values I’m going to transform.
I’ll start with my usual, -1, 0, 1. And -1³ is -1, 0³ is 0, 1³ is 1 so that’s easy. Then, I need to make a table for 10, 2/5x minus 1³ plus 10 and as I usually do, I'm going to make a substitution for this inside stuff. The 2/5 minus 1, I’m going to let u equal 2/5x minus 1. This is a more complicated transformation, so I actually need to solve for x and figure out what I need to do to these inputs to get my x values.
First I add 1 to both sides and then I have to multiply both sides by 5/2. You get 5/2 u plus 1 equals x. I can actually leave it like this because this is easy enough to work with these numbers. Take minus 1, I add 1 and I get 0, times 5/2 is 0. Start with 0, add 1, I get 1 times 5/2 is 5/2. If you start with 1 and 1 you get 2, times 5/2 is 5. That wasn’t too bad. So these are my new x values. It looks like we’ve got a shift to the right and also a horizontal stretch going on because these points were originally two units apart now they’re 5 units apart.
What about the y values? We’re going to multiply this thing by 10 and then add 10. So here are the cubes, we multiply by 10, we get -10 and add 10 we get zero. Multiply by 10 we get zero plus 10 is 10. Multiply by 10 we get 10 plus another 10 is 20. These are our 3 points, (0, 0) (5/2, 10) now on the y axis here I’m going to decide to scale this as 10 and this as 20. This will be 5 and 5/2 will be right in here. So (5/2, 10) and then (5, 20) will be up here. This is enough to get me a pretty decent graph of this cubic. It’s important to remember that this center point is the inflection point of the graph. Remember that the graph of y equals x to the 3rd power has this inflection point, this twist in it’s graph. Originally that point is (0, 0) but after the transformation it's (5/2, 10).
It’s important to remember that this point is the point where the twist occurs. Otherwise the graph’s pretty easy. Just remember make a substitution for the inside stuff, whatever that might be and solve for x. That tells you what you need to do to your table of key points in order to get your new x values. And then look at the outside of the function, you have the cube function being multiplied by 10 and then adding 10. You take these multiply by 10, add 10 and get these values. It’s as simple as that.