Question

Which function displays this end behavior?

- As x approaches negative infinity, y approaches positive infinity.
- As x approaches positive infinity, y approaches negative infinity.
a)y=−2x³−1
b)y=(x+2)³−9
c)y=3(x−1)²
d)y=−3x²+4

Answer 1

The function with the end behavior where y approaches positive infinity as x approaches negative infinity, and y approaches negative infinity as x approaches positive infinity is a third-degree polynomial with a negative leading coefficient. The correct function is a) y = -2x³ - 1.

Step-by-step explanation:

The function that displays the end behavior where as x approaches negative infinity, y approaches positive infinity, and as x approaches positive infinity, y approaches negative infinity, is a third-degree polynomial with a negative leading coefficient. Analyzing the given options:

• y = -2x³ - 1: This is a cubic function with a negative leading coefficient, which matches the described end behavior.
• y = (x+2)³ - 9: This is also a cubic function, but its leading coefficient is positive, so the end behavior does not match the description.
• y = 3(x-1)²: This is a quadratic function, and quadratic functions with positive leading coefficients do not have the end behavior described.
• y = -3x² + 4: This is a quadratic function with a negative leading coefficient; its ends go in the same direction, which contradicts the described behavior.

Based on this analysis, the correct function is a) y = -2x³ - 1.

Answer 2

The function that displays the required end behavior is a third-degree polynomial with a negative leading coefficient. The correct answer is (a) y = -2x³ - 1, as it is the only function listed that meets the criteria for the described end behavior.

Step-by-step explanation:

The function that displays the end behavior where as x approaches negative infinity, y approaches positive infinity, and as x approaches positive infinity, y approaches negative infinity is a third-degree polynomial with a negative leading coefficient. Looking at the given options, we need to find a function that satisfies both conditions. Option (a) y = -2x³ - 1 is a cubic function with a negative leading coefficient. For this type of function, as x approaches negative infinity, y approaches positive infinity, and as x approaches positive infinity, y approaches negative infinity. So this function has the described end behavior.

Comparatively, options (b) and (c) do not meet the criteria as they are either a positive cubic function or a squared function, respectively. A positive cubic function will approach positive infinity as x approaches positive infinity, and a squared function will approach positive infinity as x approaches both negative and positive infinity. Option (d), being a negative quadratic, will not have the described end behavior either, since as x approaches both negative and positive infinity, y approaches negative infinity. Therefore, the correct answer is (a) y = -2x³ - 1.