To determine which equation represents the direct variation function containing the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9)\)[/tex], follow these steps:
1. Identify the coordinates of the given points:
- Point 1: [tex]\((-8, -6)\)[/tex]
- Point 2: [tex]\( (12, 9)\)[/tex]
2. Use the slope formula to calculate the slope [tex]\( k \)[/tex] of the line passing through these points:
[tex]\[
k = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\((x_1, y_1) = (-8, -6)\)[/tex] and [tex]\((x_2, y_2) = (12, 9)\)[/tex]:
[tex]\[
k = \frac{9 - (-6)}{12 - (-8)}
\][/tex]
3. Simplify the expression to find the slope [tex]\( k \)[/tex]:
[tex]\[
k = \frac{9 + 6}{12 + 8} = \frac{15}{20} = \frac{3}{4}
\][/tex]
The slope [tex]\( k \)[/tex] of the direct variation function is [tex]\(\frac{3}{4}\)[/tex].
4. Compare the calculated slope with the given equations:
- [tex]\( y = -\frac{4}{3} x \)[/tex]: Slope is [tex]\(-\frac{4}{3}\)[/tex]
- [tex]\( y = -\frac{3}{4} x \)[/tex]: Slope is [tex]\(-\frac{3}{4}\)[/tex]
- [tex]\( y = \frac{3}{4} x \)[/tex]: Slope is [tex]\(\frac{3}{4}\)[/tex]
- [tex]\( y = \frac{4}{3} x \)[/tex]: Slope is [tex]\(\frac{4}{3}\)[/tex]
5. Identify the equation that matches the calculated slope:
- The equation [tex]\( y = \frac{3}{4} x \)[/tex] has a slope of [tex]\(\frac{3}{4}\)[/tex].
Therefore, the equation representing the direct variation function that contains the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9)\)[/tex] is:
[tex]\[
y = \frac{3}{4} x
\][/tex]
