What is the formula for direct variation where y varies directly as x^n?

In the context of direct variation, when it is stated that ( y ) varies directly as ( x^n ), it means that there is a constant ( k ) such that the relationship can be expressed as ( y = kx^n ). Here, ( k ) is a non-zero constant that represents the ratio of ( y ) to ( x^n ) for any point in the relationship.

The term "varies directly" signifies that as ( x^n ) increases, ( y ) increases proportionally, and vice versa. The exponent ( n ) indicates that the relationship involves a power of ( x ), making it distinctly different from more straightforward linear relationships, which would involve just ( x ) without an exponent.

For instance, if ( k ) is determined to be 2 and ( n ) is 3, then the equation becomes ( y = 2x^3 ). This indicates that doubling ( x^3 ) would result in doubling ( y ), illustrating the nature of direct variation.

Other options do not represent the concept of direct variation with respect to a power of ( x ):

• The formula involving ( k/x ) depicts an