Question

In the function f(x) = a(x + 2)(x-3), a and b are both integer constants and b is positive. If the end behavior of the graph of y = f(x) is positive for both very large negative values of x and very large positive values of x, what is true about a and b?

A. a is negative, and b is even.
B. a is positive, and b is even.
C. a is negative, and b is odd.
D. a is positive, and b is odd.

Answer 1

For the function f(x) = a(x + 2)(x - 3), with the graph having a positive end behavior in both directions, constant a must be positive. The constant b is irrelevant, and there appears to be a typo in the question as it does not apply to the given function.

Step-by-step explanation:

Given the function f(x) = a(x + 2)(x - 3), and the requirement that the end behavior of the graph of y = f(x) is positive for very large negative and very large positive values of x, we can determine the nature of constant a. The graph of a quadratic function opens upwards if the coefficient of the leading term is positive (positivity of a), and downwards if it's negative (negativity of a).

Since the requirement is that the end behavior is positive in both directions, a must be positive. The constant b is not applicable here because there is no 'b' in the given function. Thus, the correct answer is that a is positive, without any relevance to b. Considering the options provided, although 'b' is mentioned, it does not factor into the function given, so we only focus on the positivity of a. Given the options provided, we must assume there is a mistake in the question as none of the options accurately represent the situation where only the sign of a is relevant.