Question

a direct variation function contains the points (-8, -6) and (12, 9). which equation represents the function?\
\\( y = -\frac{4}{3}x \\)\
\\( y = -\frac{3}{4}x \\)\
\\( y = \frac{3}{4}x \\)\
\\( y = \frac{4}{3}x \\)

Explanation:

Step1: Recall direct variation form

A direct variation function has the form $y = kx$, where $k$ is the constant of variation.

Step2: Solve for $k$ using first point

Substitute $x=-8$, $y=-6$ into $y=kx$:
$$-6 = k(-8)$$
Solve for $k$:
$$k = \frac{-6}{-8} = \frac{3}{4}$$

Step3: Verify with second point

Substitute $x=12$, $k=\frac{3}{4}$ into $y=kx$:
$$y = \frac{3}{4}(12) = 9$$
This matches the given point $(12,9)$, so $k=\frac{3}{4}$ is correct.

Answer:

$\boldsymbol{y=\frac{3}{4}x}$ (the third option)