Let's analyze the function [tex]\( f(x) = |x - h| + k \)[/tex] where both [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are positive.
1. Understanding the Absolute Value Function: The absolute value function [tex]\( f(x) = |x - h| \)[/tex] forms a V-shape. The vertex of this V is at the point [tex]\( x = h \)[/tex].
2. Adding the Constant [tex]\( k \)[/tex]: By adding [tex]\( k \)[/tex] to [tex]\( |x - h| \)[/tex], the entire graph of the function [tex]\( f(x) = |x - h| \)[/tex] is shifted vertically upward by [tex]\( k \)[/tex] units.
3. Vertex and Symmetry:
- The vertex of the graph [tex]\( f(x) = |x - h| + k \)[/tex] will be at [tex]\( (h, k) \)[/tex].
- The graph will be symmetrical about the vertical line [tex]\( x = h \)[/tex].
4. Graph Characteristics:
- The graph is a V-shape.
- The vertex of the V is at the point [tex]\( (h, k) \)[/tex].
- The arms of the V extend infinitely, with one arm going downwards and to the left and the other going downwards and to the right.
- As [tex]\( x \)[/tex] moves away from [tex]\( h \)[/tex], the function values increase linearly since [tex]\( |x - h| \)[/tex] increases linearly.
Thus, the graph of the function [tex]\( f(x) = |x - h| + k \)[/tex] will be a V-shaped graph that opens upwards with its vertex at the point [tex]\( (h, k) \)[/tex], where the point of symmetry is the line [tex]\( x = h \)[/tex].
