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Graphing the Transformation y = f(x - h) - Concept

Teacher/Instructor 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.

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.

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