Transformation Functions

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yalialp06

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Why is it that when comparing y=x^2 and y=(x-1)^2 the graph of the second function is shifted to the right by 1? Intuitively, x-1 makes me think to shift the function left. I have an idea, but want to hear what others have to say.
 

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Suppose you want the y-values for both equations to be 0, 1, 4, and 25. What x-values will you use for y=x^2 and what x-values will you use for y=(x-1)^2? How do the x-values from the two functions compare to one another? Possible graphing those points will also help.
Please post back with your thoughts.
 
I just noticed that you said that you have an idea. That is wonderful! Can we please share your idea with us? As far as us giving you ideas first, that will not happen on this forum.
 
Jomo said:
Suppose you want the y-values for both equations to be 0, 1, 4, and 25. What x-values will you use for y=x^2 and what x-values will you use for y=(x-1)^2? How do the x-values from the two functions compare to one another? Possible graphing those points will also help.
Please post back with your thoughts.
Click to expand...

I notice that the x values have to be greater in the second function in order to end up with the same y values that I had in the first function. If the x values are greater than those in the previous function then the graph must shift right. I'm still trying to make sense of it though.

I just was thinking that it has something to do with the y values changing, not the x values. For example, the y values are smaller in the second function because I am squaring a smaller x value. How is this idea connected to what I noticed in the preview paragraph.
 
If one function has the point (3, 7) and a 2nd function has the point (4, 8) isn't the 2nd point shifted one place to the right?
 
yalialp06 said:
I just was thinking that it has something to do with the y values changing, not the x values. For example, the y values are smaller in the second function because I am squaring a smaller x value. How is this idea connected to what I noticed in the preview paragraph.
Click to expand...
This type of shift has nothing to do with the change in the y-value.
 
Jomo said:
If one function has the point (3, 7) and a 2nd function has the point (4, 8) isn't the 2nd point shifted one place to the right?
Click to expand...
Looking at these points, I do see that the first point shifted one place to the right because the x value increased by one. I just don't see how subtracting one from x in the actual function leads to shifting the parent function to the right.
 
I also see that since you take away 1 from x, the x values have to be greater in order to yield the same y values as in the parent function. I think that to compensate for taking 1 away from x in the second function y=(x-1)^2, I must increase the x values by 1, therefore, shifting the function right. Is that right? And yet, I still don't see it visually on the graph, the minus 1 resulting in a positive horizontal shift.
 
When I square 5 I get 25.

We are going to play a game. You need to give me a number knowing that I will square the number and want to get 25. One more rule, before squaring the number you give me I will subtract one from that number first. What number(s) will you give me?
 
The only two numbers that will work are 6 and -4. I understand that x has to equal 6 in the new function. And this x is one greater than the x on the parent function. I can understand the shift right here. What about -4?
 
Never mind. I see how -4 works too. Thank you for your help Jomo. As always, you made these mathematical ideas very clear!!
 
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