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What's the transformation of y=(x-3)^2+4?

Which way will the graph move?

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Kathye P. | Math Geek, passionate about teachingMath Geek, passionate about teaching
5.0 5.0 (150 lesson ratings) (150)

Hi, Ana.

The number inside the parentheses with the x tells us the horizontal shift; a -3 means the graph moves to the right 3.

The constant tells us the vertical movement of the graph; the +4 tells us that the graph moves up 4.

Your vertex is now at (3,4).

Not appicable to this case, but a coefficient multiplied by the (x-3) would tell us the vertical stretch of the graph.

Hope this helps!


One note - the 'vertical stretch' coefficient should be applied outside the parentheses.  Putting a coeffient inside the parentheses will also create a vertical stretch, but it will also have other [possibly undesirable] effects such as additional horizontal translation and horizontal stretching/compression (it depends on how you look at it).

You're right - I wanted to mention the information about the coefficient's effect on the graph, but it was not worded well. I'll clarify that.

Emma D. | Statics/Dynamics Online Tutor (MIT Alumna, EIT, 10+ yr exp)Statics/Dynamics Online Tutor (MIT Alumn...
5.0 5.0 (164 lesson ratings) (164)

Hi Ana, 

To visualize how the graph moves, rewrite y = (x - 3)^2 + 4 so that it is easier to compare with y = x^2. 

old: y = x^2

new: (y' - 4) = (x' - 3)^2

Now you can see that the transformation changed y to (y' - 4) and x to (x' - 3).  

y = y' - 4 ----> y' = y + 4.  This means that the new y' is the old y shifted up 4.

x = x' - 3 ----> x' = x + 3.  This means that the new x' is the old x shifted up 3.

Roman C. | Masters of Education Graduate with Mathematics ExpertiseMasters of Education Graduate with Mathe...
4.9 4.9 (344 lesson ratings) (344)

A very general rule:

If g(x) = f(x-h) + k, then the graph of g(x) is produced by shifting the graph of f(x) by h units right and k units up.

For example, if f(x) = x2then your equation is just y = f(x-3) + 4. So your function is just a shift of y = x2 by 3 units right and 4 units up.

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)

It actually depends on the original function.

If the original function is y = x2, then y = (x-3)^2 moves y = x2 to the right by 3 units, and y = (x-3)^2 + 4 moves y = (x-3)^2 up by 4 units.


Extention: Redo the problem for y=(2x-3)^2+4.