To determine the possible graph of the function [tex]\( f(x) = |x - h| + k \)[/tex] where [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive, follow the detailed steps below:
1. Understand the Form of the Function:
The given function is an absolute value function, [tex]\( f(x) = |x - h| + k \)[/tex]. An absolute value function typically produces a V-shaped graph.
2. Vertex of the Graph:
For the function [tex]\( f(x) = |x - h| + k \)[/tex]:
- The expression [tex]\( x - h \)[/tex] indicates a horizontal shift. Specifically, it shifts the standard graph of [tex]\( |x| \)[/tex] to the right by [tex]\( h \)[/tex] units.
- The constant [tex]\( k \)[/tex] adds a vertical shift. Specifically, it shifts the graph upwards by [tex]\( k \)[/tex] units.
Therefore, the vertex of the V-shaped graph will be at the point [tex]\((h, k)\)[/tex].
3. Positive Values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
- Since [tex]\( h \)[/tex] is positive, the horizontal shift is to the right of the origin (0, 0).
- Since [tex]\( k \)[/tex] is positive, the vertical shift is upwards from the origin.
Consequently, the vertex [tex]\((h, k)\)[/tex] is located in the first quadrant of the Cartesian plane.
4. Shape and Orientation:
- The V-shape will open upwards because the absolute value function [tex]\( |x - h| \)[/tex] always yields a non-negative value, and adding [tex]\( k \)[/tex] simply shifts this non-negative value upwards.
Therefore, the graph of [tex]\( f(x) = |x - h| + k \)[/tex], where [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive, will be a V-shaped graph with its vertex at the point [tex]\((h, k)\)[/tex], located in the first quadrant, and will open upwards.
Graphically, it should look something like this:
```
y
|
|
| / \
| / \
| / \
+--------------- x
```
In this sketch:
- The point marked represents the vertex [tex]\((h, k)\)[/tex].
- The graph opens upwards.
Thus, any graph of [tex]\( f(x) = |x - h| + k \)[/tex] with [tex]\( h \)[/tex] and [tex]\( k \)[/tex] being positive will have its vertex in the first quadrant and open upwards from that vertex.
