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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There are different types of **math transformation**, one of which is the type y = f(bx). This type of math transformation is a horizontal compression when b is greater than one. We can graph this math transformation by using tables to transform the original elementary function. Other important transformations include vertical shifts, horizontal shifts, and reflections.

Let's investigate another transformation. I want to know what's the transformation y=f of bx do? So let's start with the function f of x =4-x I'm going to graph that function and then I want to graph f of 4x. So let's make this our parent function it's actually pretty easy to come up with values for it and you know what the shape is going to be, it's a radical function. So I'll make this u and root 4-u, let's pick values to plug in that'll give us nice perfect squares inside the radical. So for example u=0 is a good choice because it gives you 4-0, 4 and 4 is 2.

If I want to get 1 inside here I'd pick u=3, so I'll pick 3 I get 1 inside and the square root of that is 1 and if I want to get 0 I'll pick u=4. So 4-4 is 0, 0 is 0, so these are the three points I'll use to graph it. And let me graph it right now 0 2, 3 1, 4 0 and you can see that if it's a radical function. This is it's end point so it's going to open to the left and it'll look something like this, now what does the transformation do? Well let me make a substitution, I want to graph f of 4x so let me make the substitution, first of all this is y equals the square root of 4-4x right? f of 4x is replacing the x by 4x, so I'll substitute u for 4x. Now if u equals 4x that means x equals one quarter u. That's why I take these u values and I multiply them by a quarter. And I get 0 times a quarter is 0, 3 times a quarter is three quarters and four times a quarters 1. These are my x values, and then here I'll have root 4-4x and this would be exactly the same as this because 4x is u, so 4-u exactly these values 2, 1, 0.

Alright so I'm going to plot 0, 2 three quarters 1 and 1, 0. So here is 0, 2, three quarters, 1 is here and 1, 0 is here. Now 1, 0 is the transformation of 4, 0 the old end point. So the old end point which is way out here has moved in to here, and this is what my new graph looks like. This is a horizontal compression the graph has been squeezed in to the y axis and it's a compression by a factor 1 quarter right 1 to 4, so just remember when you see the transformation f of 4x the number 4 it's greater than 1 you might expect this to be a horizontal stretch but it's actually a horizontal compression. So when you describe the transformation y=f of bx, if the b value is bigger than 1 you get a horizontal compression of the original graph by a factor of one over b.

Just like we saw here this is compression by a factor of one quarter. And those are our two graphs.