To understand what the graph of [tex]\( f(x) = |x - h| + k \)[/tex] should look like given that [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive, let’s break down the function step-by-step:
1. Basic Shape of [tex]\( f(x) = |x| \)[/tex]:
- The function [tex]\( f(x) = |x| \)[/tex] creates a V-shaped graph with its vertex at the origin (0,0). This graph has symmetry about the y-axis.
2. Horizontal Shift: [tex]\( |x - h| \)[/tex]:
- The expression [tex]\( |x - h| \)[/tex] represents a horizontal shift of the basic absolute value graph [tex]\( |x| \)[/tex].
- Because [tex]\( h \)[/tex] is positive, [tex]\( h > 0 \)[/tex], the graph of [tex]\( |x - h| \)[/tex] shifts to the right by [tex]\( h \)[/tex] units.
- Thus, if the vertex of the graph [tex]\( |x| \)[/tex] was at (0,0), the vertex of [tex]\( |x - h| \)[/tex] will now be at [tex]\( (h, 0) \)[/tex].
3. Vertical Shift: [tex]\( |x - h| + k \)[/tex]:
- Adding [tex]\( k \)[/tex], which is a positive number ([tex]\( k > 0 \)[/tex]), results in a vertical shift of the entire graph upward by [tex]\( k \)[/tex] units.
- Therefore, if the vertex of the graph [tex]\( |x - h| \)[/tex] was at [tex]\( (h, 0) \)[/tex], the vertex of [tex]\( |x - h| + k \)[/tex] will be at [tex]\( (h, k) \)[/tex].
4. Combined Transformations:
- The final function [tex]\( f(x) = |x - h| + k \)[/tex] results in a V-shaped graph.
- This V-shaped graph has been shifted right by [tex]\( h \)[/tex] units and up by [tex]\( k \)[/tex] units.
- Therefore, the vertex of the graph will be located at the point [tex]\( (h, k) \)[/tex].
Given [tex]\( h = 2 \)[/tex] and [tex]\( k = 3 \)[/tex]:
The graph of the function [tex]\( f(x) = |x - 2| + 3 \)[/tex] will have:
- A V-shaped graph.
- The vertex of this graph will be shifted to the point [tex]\( (2, 3) \)[/tex].
Thus, the graph of [tex]\( f(x) = |x - h| + k \)[/tex] will be a V-shaped graph with its vertex located at [tex]\( (2, 3) \)[/tex].
