To determine which equation represents the direct variation function that contains the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex], we need to find the slope [tex]\( m \)[/tex] of the line that passes through these points.
1. We start by using the formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Substituting the given points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex] into the slope formula:
[tex]\[
m = \frac{9 - (-6)}{12 - (-8)}
\][/tex]
3. Simplify the numerator and the denominator:
[tex]\[
m = \frac{9 + 6}{12 + 8} = \frac{15}{20}
\][/tex]
4. Simplify the fraction:
[tex]\[
m = \frac{15}{20} = \frac{3}{4}
\][/tex]
5. Since a direct variation function has the form [tex]\( y = mx \)[/tex], where [tex]\( m \)[/tex] is the slope we just calculated, we now know that the equation representing the function is:
[tex]\[
y = \frac{3}{4} x
\][/tex]
Among the provided choices, the correct equation is:
[tex]\[ y = \frac{3}{4} x \][/tex]
Therefore, the correct choice is:
[tex]\[ y = \frac{3}{4} x \][/tex]
