To determine which point also belongs to the direct variation relationship that includes the point [tex]\((4, -2)\)[/tex], we can follow these steps:
1. Understand Direct Variation:
Direct variation means that [tex]\( y \)[/tex] varies directly as [tex]\( x \)[/tex]. This can be expressed with the equation [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is a constant.
2. Find the Constant [tex]\( k \)[/tex]:
Given the point [tex]\((4, -2)\)[/tex], we plug in [tex]\( x = 4 \)[/tex] and [tex]\( y = -2 \)[/tex] into the equation [tex]\( y = kx \)[/tex]:
[tex]\[
-2 = k \cdot 4
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{-2}{4} = -0.5
\][/tex]
3. Determine if Each Point Satisfies the Equation:
We need to check if the given points satisfy the equation [tex]\( y = -0.5x \)[/tex].
- Option A: [tex]\((-4, 2)\)[/tex]
[tex]\[
y = -0.5 \cdot (-4) = 2
\][/tex]
The point [tex]\((-4, 2)\)[/tex] satisfies [tex]\( y = -0.5x \)[/tex].
- Option B: [tex]\((-4, -2)\)[/tex]
[tex]\[
y = -0.5 \cdot (-4) = 2 \quad \text{(not } y = -2 \text{)}
\][/tex]
The point [tex]\((-4, -2)\)[/tex] does not satisfy [tex]\( y = -0.5x \)[/tex].
- Option C: [tex]\((2, -4)\)[/tex]
[tex]\[
y = -0.5 \cdot 2 = -1 \quad \text{(not } y = -4 \text{)}
\][/tex]
The point [tex]\((2, -4)\)[/tex] does not satisfy [tex]\( y = -0.5x \)[/tex].
- Option D: [tex]\((-2, 4)\)[/tex]
[tex]\[
y = -0.5 \cdot (-2) = 1 \quad \text{(not } y = 4 \text{)}
\][/tex]
The point [tex]\((-2, 4)\)[/tex] does not satisfy [tex]\( y = -0.5x \)[/tex].
Based on our checks, only [tex]\((-4, 2)\)[/tex] satisfies the direct variation relationship [tex]\( y = -0.5x \)[/tex] established by the point [tex]\((4, -2)\)[/tex].
The correct answer is:
[tex]\[
\boxed{A}
\][/tex]
