Question

Which statement best describes the end behavior of

f(x)=−8x 3+2x 2+1?

Option 1: As x approaches positive infinity, f(x) approaches negative infinity.
Option 2: As x approaches positive infinity, f(x) approaches positive infinity.
Option 3: As x approaches negative infinity, f(x) approaches negative infinity.
Option 4: As x approaches negative infinity, f(x) approaches positive infinity.

Answer 1

The end behavior of the function f(x) = -8x^3 + 2x^2 + 1 is such that as x approaches positive infinity, f(x) approaches negative infinity due to the leading term with a negative coefficient.

Step-by-step explanation:

The statement that best describes the end behavior of the function f(x) = -8x^3 + 2x^2 + 1 is Option 1: As x approaches positive infinity, f(x) approaches negative infinity. This can be determined by analyzing the leading term of the polynomial, which is -8x^3. In polynomial functions, the end behavior is dominated by the term with the highest degree. Since the coefficient of this term is negative, the function will approach negative infinity as x approaches positive infinity. Additionally, as x approaches negative infinity, the function will approach positive infinity due to the odd power and negative coefficient of the leading term.

The end behavior of a polynomial function is an important concept in mathematics that describes the behavior of the function values as the input values either increase without bound (approach positive infinity) or decrease without bound (approach negative infinity).