To determine the graph of the function [tex]\(f(x) = |x - h| + k\)[/tex] given that both [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are positive, let’s analyze the function step by step.
### Understanding [tex]\(f(x) = |x - h| + k\)[/tex]
1. Absolute Value Function:
The function [tex]\(f(x) = |x - h|\)[/tex] describes an absolute value function, which is known for its V-shaped graph. The vertex of this V-shaped graph is located at point [tex]\((h, 0)\)[/tex], where the expression inside the absolute value, [tex]\(x - h\)[/tex], equals zero.
2. Translation by [tex]\(k\)[/tex]:
Adding [tex]\(k\)[/tex] to the function [tex]\(|x - h|\)[/tex] shifts the entire graph vertically upwards by [tex]\(k\)[/tex] units. Therefore, the new function [tex]\(f(x) = |x - h| + k\)[/tex] will have its vertex at [tex]\((h, k)\)[/tex].
### Position of the Vertex
Given that [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are both positive:
- [tex]\(h > 0\)[/tex]: This means the vertex is located to the right of the y-axis.
- [tex]\(k > 0\)[/tex]: This means the vertex is located above the x-axis.
The vertex of our graph is at the point [tex]\((h, k)\)[/tex], which is [tex]\((1, 1)\)[/tex].
### Shape and Location of the Graph
- The graph of [tex]\(f(x) = |x - h| + k\)[/tex] is V-shaped.
- The vertex of this V-shaped graph is situated at [tex]\((1, 1)\)[/tex], which confirms both [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are positive.
- The arms of the V open upwards from the vertex.
### Conclusion
Therefore, the graph of the function [tex]\(f(x) = |x - h| + k\)[/tex], when both [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are positive, is a V-shaped graph with the vertex located at the point [tex]\((1, 1)\)[/tex]. The vertex lies in the first quadrant of the Cartesian plane, where both x and y coordinates are positive.
