which could be the graph of $f(x) = |x - h| + k$ if $h$ and $k$ are both positive?

Answer

Explanation:

The function ( f(x) = |x - h| + k ) is a transformation of the parent absolute - value function ( y=|x|). The vertex form of an absolute - value function is ( y = |x - a|+b), where ((a,b)) is the vertex of the V - shaped graph.

Step 1: Analyze the vertex of (f(x)=|x - h|+k)

For the function (f(x)=|x - h|+k), the vertex of the graph is at the point ((h,k)). We know that (h>0) and (k > 0), so the vertex of the graph should be in the first quadrant (since both the (x) - coordinate (h) and the (y) - coordinate (k) of the vertex are positive).

Step 2: Analyze each graph

For the first graph: The vertex has a positive (x) - coordinate and a positive (y) - coordinate (it is in the first quadrant).
For the second graph: The vertex has a non - positive (y) - coordinate (since it is on or below the (x) - axis), so (k\leq0) for this graph, which does not satisfy (k > 0).
For the third graph: The vertex has a non - positive (x) - coordinate (since it is to the left of the (y) - axis), so (h\leq0) for this graph, which does not satisfy (h>0).
For the fourth graph: The graph is not a V - shaped absolute - value graph (it looks like a linear graph with a different shape), so it can be eliminated.

Answer:

The first graph (the one with the vertex in the first quadrant among the given options)