The function f is defined by f(x) = (x - 4)(x + 6). The graph of y = f(x) in the xy-plane intersects the y-axis at the…

The y-intercept of the graph of a function occurs at the point where x = 0. To find the value of k, we can substitute x = 0 into the given function: f(0) = (0 - 4)(0 + 6) = (-4)(6) = -24.

Therefore, k = -24.

The question asks for an equivalent expression for f(x) that displays the value k = -24 as a constant or coefficient. This means we are looking for the standard quadratic form, ax² + bx + c, where the constant term c is the y-intercept.

To find this form, we expand the given expression: f(x) = (x - 4)(x + 6) = x(x) + x(6) - 4(x) - 4(6) = x² + 6x - 4x - 24. Combining like terms gives f(x) = x² + 2x - 24.

This expression is equivalent to the original function, and its constant term is -24, which is the value of k. Thus, choice A is correct.

Choice B is incorrect because it is the vertex form of the function, which displays the coordinates of the vertex, (-1, -25), not the y-intercept.

Choice C is incorrect because it results from a sign error when multiplying (-4)(6) to get +24.

Choice D is incorrect because it results from an incomplete expansion of the original expression and is not equivalent to f(x).