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Why is it with function transformations that sometimes horizontal translations are the opposite of what I think they'd be?

Example: In –x^2– 4x+ 5 I would say ok so it has a reflection of the y axis, It moves right 4 units and moves up 5. But the answer is that instead of moving 4 units right, its 4 units LEFT. Why???

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Jason S. | My goal is the success of my students. Knowledge-Patience-HonestyMy goal is the success of my students. K...
4.9 4.9 (115 lesson ratings) (115)
Take y = x2 and y= (x-4).
The parabolic form is y = a(x MINUS h) PLUS k, where (h, k) is the vertex of the parabola opening up or down depending on the sign of a.
So the vertex is at (0,0) for y = x2 since y = 1(x-0)2 + 0.
Now, for y = (x-4)2, note that the vertex (which is essentially the "handle" for the parabola) is now at
(+4, 0).
So despite the (x MINUS 4) being squared, the x coordinate of the vertex was shifted from 0 to 4, which is a shift to the right of 4.
Now consider y=(x+4)2.   
That's really y = (x - (-4))2 + 0.   So the vertex was moved 4 units to the left from (0, 0) to (-4, 0).
Hope this helps.
Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
5.0 5.0 (3 lesson ratings) (3)
For quadratic functions such as this one, you should complete the square to see how the parabola is shifted and reflected. In your case,
f(x) = –x²– 4x+ 5 = -(x+2)² + 9
The +9 tells us the parabola is shifted up (positive y-direction) by 9 units relative to the origin, the +2 tells us it is shifted left by 2 units (negative x-direction), so the vertex is at (-2,9). The negative sign tells us it opens down (in the negative y-direction).
More generally, if you change a function from f(x) to f(x+A) for any positive constant A, its graph is shifted A units to the left. f(x-A) shifts the graph A units to the right. If you change f(x) to f(x)+A, it shifts it A units up, and f(x)-A is shifted A units down.
Alex V. | College doesn't make you smart. College teaches you to be resourceful.College doesn't make you smart. College ...
4.8 4.8 (73 lesson ratings) (73)
Y=x^2 is centered around origin.  
Assuming your coordinates are traditional (+ to the right and - to the left of the origin)
Analyze y=x^2 - 4x:
To bring y back to the origin, we have to add (move 4x units to the right).  
This means y moved -4x units which is 4x to the left.