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the inflection point of f(x)= -2(x-4)^3 +11 is where

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2 Answers

There are two ways to solve this problem. The quickest way is to plot the equation and view the end behaviors.
 
Since we know that the equation given... f(x) = -2(x-4)^3+11 is to the power of 3. The curve will follow the following pattern:
\                                   /
 \____        or       ____/
          \               /
           \             /
       
After analyzing the graph we can determine the end behavior. If you graph the equation you will find that it follows this pattern:
\
 \__(0,0)___
                   \
                    \
 
So as x -> -infinity, y gets bigger and bigger (up), or approaches +infinity
and as x -> +infinity, y goes down or gets bigger and bigger in the negative direction and approaches -infinity
 
 
so in all... The steps would be to
1. Determine the power of the function (in english, what number is x raised to).
2. graph the function
3. determine end behavior. (if the line goes up it approaches +infinity, if it goes down it approaches - infinity)

Comments

That makes a lot of sense, does the graph always look like that to an odd power? Additionally, what is the inflection point?  
Yes you are correct. If you go to the website 
 
 
you will see the relationship. All even power functions will look like a parabola and all odd power functions will look like a twisted parabola. (Just note that the higher the power the longer the line will be "flat" in the middle)
 
As steve mentioned above. The point of inflection is when the second derivative of the function is equal to zero [ f''(x) = 0 ]. A quick way to check is that the point of inflection is usually (not always, but typically) right in the middle of where the graph goes flat. 
i.e.
\
 \_____
      ^   \
     POI  \
 
 

Comment

The inflection point of f(x)= -2(x-4)^3 +11 is where f''(x) = 0.
 
f'(x) = -6(x-4)^2
 
f''(x) = -12(x-4)
 
f''(4) = 0, so the inflection point occurs when x = 4
 
f(4) = 11, so the inflection point is (4,11)
 
End Behavior is what happens when |x| → ∞
 
You learned in algebra 2 and/or precalculus that the End Behavior of the graph of a polynomial function depends on whether its degree is even or odd and the sign of its leading coefficient.
 
If the leading coefficient is positive then the right side end behavior is up; if the leading coefficient is negative then the right side end behavior is down. Test this with x = 10^100.
 
The left side end behavior is the same as the right side for even degree polynomial functions and opposite for odd. Test this with x = -10^100.
 
f(x) has a negative leading coefficient, so its right side end behavior will be down. f(x) is an odd (3rd) degree polynomial function so its left side end behavior will be opposite to its right side, i.e., up.