Question

select the correct answer.

if the point \\((4, -2)\\) is included in a direct variation relationship, which point also belongs in this direct variation?

\\((-4, 2)\\)
\\((-4, -2)\\)
\\((2, -4)\\)
\\((-2, 4)\\)

Explanation:

Define direct variation

A direct variation is a linear relationship represented by the equation:

$$y = kx$$

where \(k\) is a non-zero constant called the constant of variation.

Find the constant of variation

Using the given point \((4, -2)\), we substitute \(x = 4\) and \(y = -2\) into the direct variation equation:

$$-2 = k(4)$$

Solving for \(k\):

$$k = \frac{-2}{4} = -\frac{1}{2}$$

Thus, the equation for this relationship is:

$$y = -\frac{1}{2}x$$

Test the given options

We substitute the \(x\)-coordinates of the given options into our equation to see which \(y\)-coordinate matches:

• For \((4, 2)\): \(y = -\frac{1}{2}(4) = -2

eq 2\)

• For \((-4, -2)\): \(y = -\frac{1}{2}(-4) = 2

eq -2\)

• For \((2, -4)\): \(y = -\frac{1}{2}(2) = -1

eq -4\)

• For \((-2, 1)\): \(y = -\frac{1}{2}(-2) = 1\) (This point matches)

Answer:

• (A) \((4, 2)\)
• (B) \((-4, -2)\)
• (C) \((2, -4)\)
• (D) \((-2, 1)\) (Correct answer)