Question

A direct variation function contains the points (–8, –6) and (12, 9). Which equation represents the function?

Answer 1

In a direct variation, the relationship between the variables can be expressed in the form \( y = kx \), where \( k \) is a constant. We can find \( k \) using the given points.

We can take either of the points and solve for \( k \).

### Using the point (–8, –6):
\[
y = kx \\
-6 = k(-8) \\
k = \frac{-6}{-8} = \frac{6}{8} = \frac{3}{4}
\]

### Now let's check with the point (12, 9):
\[
y = kx \\
9 = k(12) \\
k = \frac{9}{12} = \frac{3}{4}
\]

Both points confirm that \( k = \frac{3}{4} \).

### Equation of the direct variation:
Now that we have the value of \( k \), we can write the equation representing the function:
\[
y = \frac{3}{4}x
\]

Thus, the equation that represents the direct variation function is:
\[
y = \frac{3}{4}x
\]