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 formula

The formula for direct variation is \( y = kx \), where \( k \) is the constant of variation. We can find \( k \) using one of the given points. Let's use the point \( (-8, -6) \). Substitute \( x = -8 \) and \( y = -6 \) into the formula:
\( -6 = k(-8) \)

Step2: Solve for \( k \)

To find \( k \), divide both sides of the equation by \( -8 \):
\( k = \frac{-6}{-8} = \frac{3}{4} \)

We can check with the other point \( (12, 9) \). Substitute \( x = 12 \) and \( k = \frac{3}{4} \) into \( y = kx \):
\( y = \frac{3}{4}(12) = 9 \), which matches the given \( y \)-value for \( x = 12 \). So the equation is \( y = \frac{3}{4}x \).

Answer:

\( y = \frac{3}{4}x \) (corresponding to the option "y = 3/4 x")