Question

How do you find the end behavior of \[{x^3} - 4{x^2} + 7\]?

Answer 1

Hint: Here the given question need us to get the value of the function or the behavior of the function at positive and negative extremities, which means when we put the value as positive infinity and negative infinity then the question need us to check the value of function for these extremities.

Complete step by step solution:
The given question is \[{x^3} - 4{x^2} + 7\]
Here we have to check the value of the function for positive infinity and negative infinity:
On solving first for positive infinity, we see that the variable “x” is here two times, the first time it is in cubic form which means the positive infinity will remain as positive and negative will become negative, now for the second time “x” is in square form hence sign of infinity will not affect the value.
Now plotting everything in mathematical form we get:
\[
   \Rightarrow for\,x = + \infty ,\,we\,get: \\
   \Rightarrow {x^3} = {\infty ^3} = \infty \\
   \Rightarrow - 4{x^2} = - 4({\infty ^2}) = - \infty \\
   \Rightarrow 7 = 7 \\
   \Rightarrow (\infty ) + ( - \infty ) + 7 = 7 \\
  hence\,for\,x \to \infty ,\,value\, \to 7 \\
 \]
Now similarly we have to solve for the negative extremity that is for “x” as minus infinity, on solving we get:
\[
   \Rightarrow for\,x = - \infty ,\,we\,get: \\
   \Rightarrow {x^3} = - {\infty ^3} = - \infty \\
   \Rightarrow - 4{x^2} = - 4( - {\infty ^2}) = - \infty \\
   \Rightarrow 7 = 7 \\
   \Rightarrow ( - \infty ) + ( - \infty ) + 7 = - \infty \\
  hence\,for\,x \to - \infty ,\,value\, \to - \infty \\
 \]
Here values at extremities are seven for positive extremity and minus infinity for negative extremity.


Note: The given question can also be solved by plotting the curve for the given function, for which we need to expand the expression and then by assuming different values of “x” we can assume it for a function say “f(x)”, now with the help of coordinates we can plot the graph and see the behavior at extremities.