What happens to a hyperbola if the value of k increases?

When the value of ( k ) increases in the standard equation of a hyperbola, it influences the distance between the branches of the hyperbola from the axes. In a typical hyperbola described by the equation (\frac{(y-k)^2}{a^2} - \frac{x^2}{b^2} = 1) or similar forms, the term ( k ) largely affects the vertical positioning of the hyperbola, while the parameters ( a ) and ( b ) influence the shape and spread.

As ( k ) increases, specifically for the vertical hyperbola, the branches move higher up on the coordinate plane, thus increasing the vertical distance from the x-axis. Similarly, for horizontal hyperbolas affected by ( k ), a higher ( k ) value results in the branches moving further away from the y-axis. Essentially, an increase in ( k ) translates to a greater separation between the branches and the axis they are associated with, making them spread further apart.

This observation demonstrates that as ( k ) increases, the branches indeed move further away from the axes, confirming the correctness of that response. The other choices suggest either a collapse of the hyperbola or a transformation into