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

A direct variation function has the form \( y = kx \), where \( k \) is the constant of variation. To find \( k \), we can use the formula \( k=\frac{y}{x} \) (since \( y = kx\Rightarrow k=\frac{y}{x} \)) for a point \((x,y)\) on the function.

Step2: Calculate \( k \) using the first point \((-8, -6)\)

Substitute \( x=-8 \) and \( y = -6 \) into \( k=\frac{y}{x} \). So \( k=\frac{-6}{-8}=\frac{6}{8}=\frac{3}{4} \).

Step3: Verify with the second point \((12, 9)\)

Substitute \( x = 12 \) and \( y=9 \) into \( k=\frac{y}{x} \). Then \( k=\frac{9}{12}=\frac{3}{4} \). Both points give the same \( k \), so the equation is \( y=\frac{3}{4}x \).

Answer:

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