Let's break down the steps needed to find the equation of the direct variation function that contains the given points [tex]\( (-8, -6) \)[/tex] and [tex]\( (12, 9) \)[/tex].
### Step 1: Understand Direct Variation
A direct variation function can be written in the form [tex]\( y = kx \)[/tex], where [tex]\( k \)[/tex] is the constant of variation (slope).
### Step 2: Find the Slope [tex]\( k \)[/tex]
The slope [tex]\( k \)[/tex] for a direct variation function passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ k = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
### Step 3: Plug in the Given Points to Find [tex]\( k \)[/tex]
We have the points [tex]\((-8, -6)\)[/tex] and [tex]\( (12, 9) \)[/tex]. Using these points:
[tex]\[ x_1 = -8, \, y_1 = -6, \, x_2 = 12, \, y_2 = 9 \][/tex]
Calculate the slope:
[tex]\[ k = \frac{9 - (-6)}{12 - (-8)} \][/tex]
[tex]\[ k = \frac{9 + 6}{12 + 8} \][/tex]
[tex]\[ k = \frac{15}{20} \][/tex]
[tex]\[ k = \frac{3}{4} \][/tex]
So, the slope [tex]\( k \)[/tex] is [tex]\( \frac{3}{4} \)[/tex].
### Step 4: Write the Equation of the Direct Variation Function
Using the value of [tex]\( k \)[/tex], we substitute it into the direct variation form [tex]\( y = kx \)[/tex]:
[tex]\[ y = \frac{3}{4} x \][/tex]
### Step 5: Check the Possible Answers
We need to identify which of the given options matches our derived equation:
- [tex]\( y = \frac{4}{3} x \)[/tex]
- [tex]\( y = \frac{3}{4} x \)[/tex]
- [tex]\( y = \frac{3}{4} x \)[/tex] (duplicate)
- [tex]\( y = \frac{4}{3} x \)[/tex] (duplicate)
Clearly, the correct equation that matches our solution is:
[tex]\[ y = \frac{3}{4} x \][/tex]
Thus, 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]
