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 graphing transformation, one of which is subtraction a constant from the independent variable. This type of **graphing transformation** can be written as y = f(x - h). For this graphing transformation, we shift the graph horizontally by h units. We should also know how to recognize vertical shifts and scaling, reflections, and horizontal compression.

I want to talk about the transformation y equals f of x minus h. And to understand what kind of transformation this gives us, let's look at an example where I graph three functions that are all related by this transformation. Notice in these two functions, I've replaced the the x and root x by something else. Here x+4, here x-1.

Let's start by plotting some key points for y equals root x and I've made the substitution u u for x for a reason that you'll see in a moment but let's just write down some numbers here.

Now I like to use perfect squares nice numbers that are easy to take the square root of. So 0, 1 and 4 are pretty nice numbers. And the square roots are 0, 1 and 2.And so we can plot the square root of x really easily just using these three points. 0 0, 1 1 and 4 2 and here's the square root of x. Alright, now let's plot this function y equals root x plus 4. And here I make the substitution u=x+4 and that means x=u-4. What that tells me, excuse me, is that I take the u values from the square root graph and I subtract 4 from them to get my x values for this graph.

So subtract 4, I get -4, subtract 4 I get -3, subtract 4 I get 0. But nothing happens to the y values. This is the square root of u so I just copy these y values over, 0, 1 and 2 and when we plot these three points -4 0, -3 1 and 0 2. -4 0, -3 1 and 0 2 is right here. So that's what happened. This graph has basically shifted to the left four units. Note I had x+4 and the graph has shifted to the left four units. The +4 you might think shifts the graph to the right. It actually shifts the graph to the left. It's the opposite of what you think.

Let's take a look at another example. y equals root x minus one. I'll make the same substitution u=x-1 and I add one to both sides u+1=x. So my x values I get by adding 1 to my u values here. So I add 1 and I get 1, I add 1, I get 2, add 1 I get 5, but this is just the square root of u so nothing happens to the u values I just copy them over. 0, 1 and 2 and here are my points 1 0, 2 1 and 5 2. 1 0, 2 1 and 3, 4, 5, 2. And you could see that the square root of, this is y equals root u x+x-1. The x-1 indicate to shift to the right one unit. Again it's counter intuitive. The x-1 you might think shifts the graph to the left but it shifts it to the right.

So let's just review really quickly what this transformation does. y equals half of x x-h is a horizontal shift. If each is positive it shifts the graph to the right. Like when h was one, we had x-1 the graph was shifted to the right one unit. In this instance you could think of h as being -4. It's like x minus -4 the graph shifts to the left four units. That's how horizontal translation works.