The statement "
y
varies directly as
x
," means that when
x
increases,
y
increases by the same factor. In other words,
y
and
x
always have the same ratio:
= k |
y = kx |
Since k is constant (the same for every point), we can find k when given any point by dividing the y-coordinate by the x-coordinate. For example, if y varies directly as x , and y = 6 when x = 2 , the constant of variation is k = = 3 . Thus, the equation describing this direct variation is y = 3x .
Example 1: If
y
varies directly as
x
, and
x = 12
when
y = 9
, what is the equation that describes this direct variation?
k = =
y =
x
Example 2: If
y
varies directly as
x
, and the constant of variation is
k =
, what is
y
when
x = 9
?
y =
x = (9) = 15
As previously stated, k is constant for every point; i.e., the ratio between the y -coordinate of a point and the x -coordinate of a point is constant. Thus, given any two points (x _{1}, y _{1}) and (x _{2}, y _{2}) that satisfy the equation, = k and = k . Consequently, = for any two points that satisfy the equation.
Example 3: If
y
varies directly as
x
, and
y = 15
when
x = 10
, then what is
y
when
x = 6
?
=
=
6() = y
y = 9
An equation of the form y = kx can be thought of as an equation of the form y = mx + b where m = k and b = 0 . Thus, a direct variation equation is an equation in slope- intercept form which passes through (0, 0) and has a slope equal to the constant of variation.
Therefore, to graph a direct variation equation, start at (0, 0) and then proceed as you would in graphing a slope. Or, if you know one point, draw a straight line between (0, 0) and that point, and extend the line on both sides.
Example 4: y varies directly as x . If the constant of variation is , graph the line which represents the variation, and write an equation that describes the variation.
To calculate the constant of variation, given a graph of direct variation, simply calculate the slope.