The function f is defined by f(x) = x^2 - 2x - 8. Which of the following equivalent forms of f(x) displays the x-interc…

Correct answer: A. f(x) = (x - 4)(x + 2)

The question asks for an equivalent form of the given function that displays the x-intercepts as constants or coefficients. The x-intercepts are the values of x for which f(x) = 0. The factored form of a quadratic function, f(x) = a(x-p)(x-q), displays the x-intercepts, p and q, as constants. To find the factored form of f(x) = x^2 - 2x - 8, we need to find two numbers that have a product of -8 and a sum of -2. These two numbers are -4 and 2. Therefore, the factored form of the function is f(x) = (x - 4)(x + 2). In this form, setting f(x) = 0 gives (x - 4)(x + 2) = 0, which has solutions x = 4 and x = -2. These are the x-intercepts. Choice A is correct because it is the factored form and displays the numbers related to the x-intercepts (4 and -2) as constants. Choice B is incorrect. Expanding (x + 4)(x - 2) gives x^2 + 2x - 8, which is not equivalent to the original function. This results from a sign error during factoring. Choice C is the vertex form of the function. It is an equivalent expression, but it displays the coordinates of the vertex, (1, -9), not the x-intercepts. Choice D is an equivalent expression, but it does not display the x-intercepts as constants or coefficients. Setting x(x - 2) - 8 = 0 does not immediately reveal the solutions.