To determine which equation represents the direct variation function containing the points [tex]\((-9, -3)\)[/tex] and [tex]\((-12, -4)\)[/tex], we can follow these steps:
### Step 1: Identify the coordinates of the points
The given points are [tex]\((-9, -3)\)[/tex] and [tex]\((-12, -4)\)[/tex].
### Step 2: Calculate the slope of the direct variation
The formula to find the slope [tex]\(k\)[/tex] in direct variation from two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ k = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the coordinates of the given points:
[tex]\[ k = \frac{-4 - (-3)}{-12 - (-9)} = \frac{-4 + 3}{-12 + 9} = \frac{-1}{-3} = \frac{1}{3} \][/tex]
So, the slope [tex]\(k\)[/tex] of the direct variation is [tex]\(\frac{1}{3}\)[/tex].
### Step 3: Form the equation of the direct variation
The direct variation equation is of the form:
[tex]\[ y = kx \][/tex]
With [tex]\(k = \frac{1}{3}\)[/tex], the equation becomes:
[tex]\[ y = \frac{1}{3}x \][/tex]
### Step 4: Select the correct option
Looking at the given options:
1. [tex]\( y = -3x \)[/tex]
2. [tex]\( y = -\frac{x}{3} \)[/tex]
3. [tex]\( y = \frac{x}{3} \)[/tex]
4. [tex]\( y = 3x \)[/tex]
The equation [tex]\( y = \frac{1}{3}x \)[/tex] matches with option 3:
[tex]\[ y = \frac{x}{3} \][/tex]
### Conclusion
The equation that represents the direct variation function containing the points [tex]\((-9, -3)\)[/tex] and [tex]\((-12, -4)\)[/tex] is:
[tex]\[ \boxed{y = \frac{x}{3}} \][/tex]
