To determine which equation could represent the quadratic function [tex]\( f \)[/tex] given its x-intercepts, we need to follow these steps:
1. Identify the x-intercepts: The x-intercepts of the function are [tex]\((-7,0)\)[/tex] and [tex]\((-4,0)\)[/tex]. These intercepts tell us that the function will become zero when [tex]\( x \)[/tex] is [tex]\(-7\)[/tex] or [tex]\(-4\)[/tex].
2. Form the factors: Given the x-intercepts, the quadratic function can be written in factored form as:
[tex]\[
f(x) = a (x + 7)(x + 4)
\][/tex]
Here, [tex]\( a \)[/tex] is a constant that can stretch or compress the graph vertically but does not change the x-intercepts.
3. Check each given option:
- Option A. [tex]\( f(x) = -\frac{1}{2}(x - 7)(x + 4) \)[/tex]
- The factors [tex]\( (x-7) \)[/tex] and [tex]\( (x+4) \)[/tex] suggest intercepts at [tex]\( x = 7 \)[/tex] and [tex]\( x = -4 \)[/tex], which is not consistent with the given intercepts [tex]\((-7, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex]. Therefore, this option is incorrect.
- Option B. [tex]\( f(x) = (x - 7)(x - 4) \)[/tex]
- The factors [tex]\( (x-7) \)[/tex] and [tex]\( (x-4) \)[/tex] suggest intercepts at [tex]\( x = 7 \)[/tex] and [tex]\( x = 4 \)[/tex], which do not match the given intercepts. This option is incorrect as well.
- Option C. [tex]\( f(x) = 2(x + 7)(x - 4) \)[/tex]
- The factors [tex]\( (x+7) \)[/tex] and [tex]\( (x-4) \)[/tex] suggest intercepts at [tex]\( x = -7 \)[/tex] and [tex]\( x = 4 \)[/tex]. This option does not have the correct intercepts and is therefore incorrect.
- Option D. [tex]\( f(x) = -3(x + 7)(x + 4) \)[/tex]
- The factors [tex]\( (x+7) \)[/tex] and [tex]\( (x+4) \)[/tex] suggest intercepts at [tex]\( x = -7 \)[/tex] and [tex]\( x = -4 \)[/tex], which are consistent with the given intercepts. The constant [tex]\(-3\)[/tex] is a vertical stretch factor and does not alter the x-intercepts. This option is correct.
Therefore, the equation that could represent the function [tex]\( f \)[/tex] with x-intercepts [tex]\((-7, 0)\)[/tex] and [tex]\((-4, 0)\)[/tex] is:
[tex]\[
\boxed{D. \, f(x) = -3(x + 7)(x + 4)}
\][/tex]
