Question

Describe the end behavior of the function:

f(x) = x^3 + 4x^2 + 4x
a) As x approaches negative infinity, y approaches positive infinity.
b) As x approaches 0, y approaches 0.
c) As x approaches negative infinity, y approaches 0.
d) As x approaches 2, y approaches 0.

Answer 1

The end behavior for the cubic function f(x) = x^3 + 4x^2 + 4x is that as x approaches negative infinity, y approaches negative infinity, and as x approaches positive infinity, y approaches positive infinity.

Step-by-step explanation:

The end behavior of a polynomial function can often be determined by looking at the leading term, which is the term with the highest power of x. For the function f(x) = x^3 + 4x^2 + 4x, the leading term is x^3. Because the power of x is odd and the coefficient is positive, the end behavior is as follows:

• As x approaches negative infinity, y approaches negative infinity.
• As x approaches positive infinity, y approaches positive infinity.

Therefore, the correct description of the end behavior for this function is not listed among the options provided by the student. None of the options accurately describe the end behavior of f(x). However, for learning purposes, the closest incorrect option to the actual behavior of the function would be option (a), although it states y approaches positive infinity as x approaches negative infinity, which is incorrect

Answer 2

The end behavior of the function f(x) = x^3 + 4x^2 + 4x is described by option (a): As x approaches negative infinity, y approaches positive infinity.

The answer is option ⇒A

Step-by-step explanation:

The end behavior of a function can be determined by examining the leading term of the polynomial. In the given function, f(x) = x^3 + 4x^2 + 4x, the leading term is x^3.

For large values of x (both positive and negative), the value of x^3 will dominate the other terms. Since x^3 approaches positive infinity as x approaches positive infinity, and x^3 approaches negative infinity as x approaches negative infinity, the end behavior of the function is described by option (a): As x approaches negative infinity, y approaches positive infinity.

The answer is option ⇒A