Exponential Parameters: the 'k' Value

Definition
In the exponential function [f(x) = a \cdot b^{(x - h)} + k] the constant ( k ) represents a vertical translation (vertical shift) of the graph.

Effect of Changing ( k )
If ( k > 0 ): The graph of ( f(x) ) shifts upward by ( k ) units.
If ( k < 0 ): The graph of ( f(x) ) shifts downward by ( |k| ) units.
The shape, growth rate, and horizontal position of the graph do not change — only its vertical position does.

Impact on the Horizontal Asymptote
The horizontal asymptote of ( f(x) ) is always [y = k]
This means the entire graph moves up or down as ( k ) changes.

Common Misconceptions
Confusing ( k ) with ( h ):
( h ) moves the graph left or right (horizontal shift).
( k ) moves the graph up or down (vertical shift).
Forgetting that the asymptote also shifts with ( k ).
Assuming ( k ) affects the growth rate — it doesn’t; that’s controlled by ( b ).