Question

Which factor should be included in the function below so that the graph of the function is increasing as x approaches negative infinity and decreasing as x approaches positive infinity?

f(x) = (x + 4)(x - 5)

A. -0.8
B. -3x
C. (x + 5)
D. (2x + 1)
E. (3x² + 5)

Answer 1

To ensure the function f(x) is increasing as x approaches negative infinity and decreasing as x approaches positive infinity, the correct factor to include is option B: -3x. This factor provides the necessary negative coefficient without changing the function to a higher degree polynomial.

Step-by-step explanation:

To determine which factor should be included in the function f(x) = (x + 4)(x - 5) so that the graph is increasing as x approaches negative infinity and decreasing as x approaches positive infinity, we need a factor that introduces an overall negative sign in the polynomial function. The function is already a quadratic with roots at x = -4 and x = 5, implying it opens upwards. Adding a negative coefficient will flip the parabola so it opens downwards, which aligns with the requirements.

Looking at the options, the factor that includes a negative coefficient and does not change the degree of the polynomial (to remain a parabola) is option B, -3x. Adding this factor gets us f(x) = -3x(x + 4)(x - 5), which satisfies the condition of increasing as x approaches negative infinity and decreasing as x approaches positive infinity.