Which way will the graph move?
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!
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.
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.
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.
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Extention: Redo the problem for y=(2x-3)^2+4.
Comments
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.