Question 1196152
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We don't have to restrict f(x) to be of the form  ax^3 + bx^2 + cx + d. The function f(x) can be anything really.


All that matters is that r is a root of f(x), i.e. f(r) = 0
The input x = r leads to the output f(x) = 0
We say that r is a zero of f(x), which in my opinion can be misleading because r itself may be zero or nonzero. 


So if we wanted to see if x = r-h is a root, then,
f(x+h) = f( (r-h)+h )
f(x+h) = f( (r-h)+h )
f(x+h) = f( r + (-h+h) )
f(x+h) = f( r + 0 )
f(x+h) = f( r )
f(x+h) = 0


Therefore, if f(r) = 0, then r-h has been confirmed to be a root of f(x+h) = 0
I.e. if r is a root of f(x), then r-h is a root of f(x+h).


What's really happening is the move from x to x+h shifts the xy  axis h units to the right. 
This gives the illusion the f(x) curve shifts h units to the left. 
That's why the root x = r shifts h units to the left to get to x = r-h


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An example:
{{{f(x) = 2^x - 32}}}
we have x = 5 as a root because
{{{f(5) = 2^5 - 32 = 32-32 = 0}}}
So r = 5 in this case


Now consider shifting the xy axis h = 3 units to the right
The old input x is now x+h, in this case x+3


Shifting the xy axis 3 units to the right gives the illusion the exponential curve shifts 3 units to the left.
So we'll be shifting the old root (r = 5) to its new spot of r-h = 5-3 = 2 which is also a shift of 3 units to the left.


Then notice the following:
{{{f(x) = 2^x - 32}}}


{{{f(x+h) = 2^(x+h) - 32}}}


{{{f(x+3) = 2^(x+3) - 32}}} Plug in h = 3


{{{f(r-h+3) = 2^(r-h+3) - 32}}} Replace every x with r-h


{{{f(5-3+3) = 2^(5-3+3) - 32}}} Plug in r = 5 and h = 3


{{{f(5) = 2^(5) - 32}}}


{{{f(5) = 32 - 32}}}


{{{f(5) = 0}}}
We arrive back at the same conclusion as before, except this time we have a specific numeric example rather than a more vague algebraic statement.


Of course one numeric example is not enough to prove a theorem entirely (we would have to check the infinitely many numbers on the number line which isn't feasible). It's sole purpose is to help illustrate what's going on. I recommend graphing out your favorite function to get practice with this. Desmos is a good free tool to do such a thing.


Once again it could be of the form  ax^3 + bx^2 + cx + d, but it doesn't have to be. 
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