To determine the equation of a direct variation function that contains the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex], we need to find the slope of the line that passes through these points and then express the function in the form [tex]\( y = mx \)[/tex].
Let's go through the step-by-step process:
1. Identify the coordinates:
- Point 1: [tex]\((x_1, y_1) = (-8, -6)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (12, 9)\)[/tex]
2. Calculate the slope ([tex]\(m\)[/tex]) of the line:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the values:
[tex]\[
m = \frac{9 - (-6)}{12 - (-8)} = \frac{9 + 6}{12 + 8} = \frac{15}{20} = \frac{3}{4}
\][/tex]
3. Write the equation of the direct variation function:
A direct variation function can be written as [tex]\( y = mx \)[/tex], where [tex]\( m \)[/tex] is the slope.
Given that [tex]\( m = \frac{3}{4} \)[/tex], we can write:
[tex]\[
y = \frac{3}{4} x
\][/tex]
Therefore, the equation that represents the direct variation function containing the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex] is:
[tex]\[
\boxed{y = \frac{3}{4} x}
\][/tex]
