Question

A direct variation function contains the points (–9, –3) and (–12, –4). Which equation represents the function?

Answer 1

A direct variation equation is one that requires y varies directly as x and looks like this in equation form:

$\frac{y}{x} =k$

where k is the constant of variation. If we solve this for y, we have y = kx, which happens to be a linear function... a line. k here, then, serves as the slope. So what we are given as points on a direct variation function are actually points on a line. The equation for this requires that we find the slope and then rewrite the formula accordingly. First the slope:

$m(k)=\frac{-4-(-3)}{-12-(-9)}=\frac{-4+3}{-12+9}=\frac{1}{3}$

Now we need to write the equation by using one of the points' coordinates. I picked the first point that has an x coordinate of -9 and a y coordinate of -3. Fitting those into the slope-intercept form of a line,

$-3=\frac{1}{3}(-9)+b$

which simplifies to

-3 = -3 + b and b = 0. That means that the equation of direct variation is

$y=\frac{1}{3}x+0$ or just

$y=\frac{1}{3}x$