How do you find the end behavior of #f(x) = - 6x^2 + 14x + 21#?

Question

Benjamin Achtenberg · Accepted Answer

It would fall to the left and fall to the right

End behaviour of a function is decided by the sign of its leading term. The leading term here is #-6x^2#. It would always be negative whether x is positive or negative. That means, as x approaches +#oo# or -#oo#, f(x) would always be negative. So, f(x) would fall to the left and also fall to the right in the interval (-#oo, +oo#)

Elijah Allen · Answer

undefined To find the end behavior of the function $ f(x) = -6x^2 + 14x + 21 $:

1. Examine the leading term: $ -6x^2 $.
2. As $ x $ approaches positive infinity, the term $ -6x^2 $ dominates, and since $ x^2 $ grows much faster than $ x $, $ f(x) $ approaches negative infinity.
3. As $ x $ approaches negative infinity, the term $ -6x^2 $ still dominates, and since $ x^2 $ grows much faster than $ x $, $ f(x) $ also approaches negative infinity.

Therefore, the end behavior of $ f(x) = -6x^2 + 14x + 21 $ is as follows:
- As $ x $ approaches positive infinity, $ f(x) $ approaches negative infinity.
- As $ x $ approaches negative infinity, $ f(x) $ also approaches negative infinity.

Ariana Alvarez · Answer

undefined To find the end behavior of the function f(x) = -6x^2 + 14x + 21, we look at the leading term of the polynomial, which is -6x^2. The leading coefficient is -6, and the degree of the polynomial is 2.

Since the leading coefficient is negative and the degree of the polynomial is even, the end behavior of the function is as follows:
- As x approaches negative infinity, the value of f(x) approaches negative infinity.
- As x approaches positive infinity, the value of f(x) approaches negative infinity.

Therefore, the end behavior of the function f(x) = -6x^2 + 14x + 21 is that it decreases without bound as x approaches negative or positive infinity.