Final answer:
Option 'd. y = 5x' represents a direct variation, as it is in the form of y = kx with a constant of variation k=5 and no y-intercept, indicating the line passes through the origin.
Step-by-step explanation:
In mathematics, particularly algebra, direct variation occurs when two variables are related in such a way that the ratio of their values remains constant. If 'y' varies directly with 'x,' the equation can be represented as y = kx, where 'k' is a non-zero constant known as the constant of variation. Among the given options:
- 'a. y = 3x-2' is not a direct variation because it has a y-intercept other than zero.
- 'b. f(x) = 1/2 x + 6' also has a y-intercept that is not zero, thus it's not a direct variation.
- 'c. y = 5' represents a horizontal line, which is a special case of a linear equation where y does not vary with x.
- 'd. y = 5x' is a direct variation because it is in the form y = kx, where the constant of variation k is 5 and there is no y-intercept, which means the line passes through the origin.
Therefore, the equation which represents a direct variation is 'd. y = 5x'.