The graph of f (in blue) is translated a whole number of units horizontally and vertically to obtain the graph of h (in red).
The function f is defined by f(x)=−|x|.
Write down the expression for h(x).

Question

The graph of f (in blue) is translated a whole number of units horizontally and vertically to obtain the graph of h (in red).
The function f is defined by f(x)=−|x|. 
Write down the expression for h(x).

anonymous · Answer

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zepdrix · Answer

So we start with the uhhh, blue :)
Red is opening in the same direction, downward.
So we don't have a reflection, that's nice.

Only translations. So just pay attention to that `vertex` point, ya?
How far did we move right and down?

anonymous · Answer

um 2 right and 4 down?

zepdrix · Answer

k good.

zepdrix · Answer

In order to shift our vertex point 2 to the right, we want to replace our x with x-2.
That gives us a new "zero" location.

$$\Large\rm f(x)=-|x|$$So we're creating this new function:$$\Large\rm y=-|x-2|$$Understand why it's x-2 and not x+2?

anonymous · Answer

oh okay I understand So is that the new expression?

zepdrix · Answer

no, that only takes care of the `horizontal` movement :)
We've done this so far:|dw:1430641339563:dw|We've made it to the green so far, by creating a function that shifts the blue to the right.

anonymous · Answer

okay

zepdrix · Answer

We want to do the same with the y.
We want to move 4 down, yes?
So we'll replace y with y+4.
-4+4 gives us a new "zero" location.

$$\Large\rm y-4=-|x-2|$$You're always doing the opposite of what seems correct.
We needed to move right 2, so we subtracted 2 from x.
We needed to move down 4, so we added 4 to y.

zepdrix · Answer

We need this in a more standard form though,
so we'll add 4 to each side, ya?

zepdrix · Answer

$$\Large\rm y\cancel{-4+4}=-|x-2|+4$$
$$\Large\rm y=-|x-2|+4$$

zepdrix · Answer

And at that point, it's ok to call this h(x) instead of y since we've completed all of our steps! :)
$$\Large\rm h(x)=-|x-2|+4$$

anonymous · Answer

okay thank you so much I'll use your examples to help me on my next problems.

zepdrix · Answer

Oh sorry sorry typo there.

zepdrix · Answer

We should've had y+4 on the left side, so we needed to `subtract` 4 from each side.

zepdrix · Answer

$$\Large\rm h(x)=-|x-2|-4$$

anonymous · Answer

okay thanks :) This made me understand it more because I was confused.

zepdrix · Answer

Ok cool c: translations can be a little weird hehe