If P(4,-5) is a point on the graph of function y= f(x),find the corresponding
point on the graph of y=2f(x-6). 
I understand that where ever I see an x, I'm supposed to substitute the
function. I am confused in how to deal with the Point(4, -5)
Thanks for your consideration.


The object is to figure out what things have been done to f(x) in order to get
2f(x-6)

Two things have been done.  

1.  First f(x) has been multiplied by 2 to give 2f(x). That stretches all the
points above the x-axis twice as high, and all points below the x-axis twice as
low.

2.  Then after doing that, x has been replaced by x-6 in 2f(x), to get 2f(x-6).
That shifts all points 6 units right.

Now let's see what has been done to the point (4,-5)

Step 1. has stretched the point (4,-5), which is below the x-axis, twice as low
below the x-axis as it is. That is to say, (4,-5) has been stretched down to
the point (4,-10) by step 1.

Then Step 2. has shifted the point (4,-10) right by 6 units to the point
(10,-10)

So that's the answer (10,-10) 

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Notice that we could have reversed 1 and 2 above.

Let's swap the two things that have been done above.  

1. x has been replaced by x-6 in f(x), to get f(x-6). That shifts all points 6
units right.  

2.  Then after doing that, f(x-6) has been multiplied by 2 to give 2f(x-6).
That stretches all the points above the x-axis twice as high, and all points
below the x-axis twice as low.

Now let's see what has been done to the point (4,-5)

Step 1.  has shifted the point (4,-5) right by 6 units to the point
 (10,-5)

Then Step 2 has stretched the point (10,-5), which is below the x-axis, twice
as low below the x-axis as it is. That is to say, (10,-5) has been stretched
down to the point (10,-10) by step 1.

So either way you get (10,-10) as the point on y = 2f(x-6) corresponding to the
point (4,-5) on f(x).

Edwin