To determine which equation, when graphed, has [tex]\( x \)[/tex]-intercepts at [tex]\((2,0)\)[/tex] and [tex]\((4,0)\)[/tex] and a [tex]\( y \)[/tex]-intercept of [tex]\((0,-16)\)[/tex], follow these steps:
First, understand that the [tex]\( x \)[/tex]-intercepts occur where the function [tex]\( f(x) \)[/tex] crosses the x-axis, i.e., where [tex]\( f(x) = 0 \)[/tex]. For an intercept at [tex]\( x = a \)[/tex], [tex]\( (x-a) \)[/tex] is a factor of the function.
Given [tex]\( x \)[/tex]-intercepts at [tex]\((2,0)\)[/tex] and [tex]\((4,0)\)[/tex], our function must include the factors [tex]\((x-2)\)[/tex] and [tex]\((x-4)\)[/tex].
Now evaluate the given options:
1. [tex]\( f(x) = -(x-2)(x-4) \)[/tex]
2. [tex]\( f(x) = -(x+2)(x+4) \)[/tex]
3. [tex]\( f(x) = -2(x-2)(x-4) \)[/tex]
4. [tex]\( f(x) = -2(x+2)(x+4) \)[/tex]
Options 1 and 3 contain the factors [tex]\((x-2)\)[/tex] and [tex]\((x-4)\)[/tex], making them possible candidates for the correct function, while options 2 and 4 can be discarded since they include incorrect factors, [tex]\((x+2)\)[/tex] and [tex]\((x+4)\)[/tex].
Next, check which of these functions has the correct [tex]\( y \)[/tex]-intercept. The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into each candidate function and see if the result is [tex]\(-16\)[/tex]:
1. [tex]\( f(x) = -(x-2)(x-4) \)[/tex]
[tex]\[ f(0) = -(0-2)(0-4) = -(-2)(-4) = -8 \][/tex]
which is not [tex]\(-16\)[/tex].
3. [tex]\( f(x) = -2(x-2)(x-4) \)[/tex]
[tex]\[ f(0) = -2(0-2)(0-4) = -2(-2)(-4) = -2 \times 8 = -16 \][/tex]
which matches the given [tex]\( y \)[/tex]-intercept [tex]\((0, -16)\)[/tex].
Therefore, based on checking both the [tex]\( x \)[/tex]-intercepts and the [tex]\( y \)[/tex]-intercept, the equation that satisfies all given conditions is:
[tex]\[ f(x) = -2(x-2)(x-4) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{f(x) = -2(x-2)(x-4)} \][/tex]
