To solve this question, we need to understand how the vertex of an absolute value function in standard form is determined. The standard form of an absolute value function is given by:
[tex]\[ f(x) = a|x - h| + k \][/tex]
In this form:
- [tex]\(a\)[/tex] is a constant that determines the steepness and direction (upwards or downwards) of the graph.
- [tex]\(h\)[/tex] is the horizontal translation of the vertex.
- [tex]\(k\)[/tex] is the vertical translation of the vertex.
The point [tex]\((h, k)\)[/tex] represents the vertex of the absolute value function.
Here is a step-by-step explanation:
1. Identify [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
- [tex]\(h\)[/tex]: This is the value that shifts the graph horizontally. If [tex]\(h\)[/tex] is positive, the shift is to the right; if [tex]\(h\)[/tex] is negative, the shift is to the left.
- [tex]\(k\)[/tex]: This is the value that shifts the graph vertically. If [tex]\(k\)[/tex] is positive, the shift is upward; if [tex]\(k\)[/tex] is negative, the shift is downward.
2. Determine the vertex:
- The vertex of the absolute value function [tex]\( f(x) = a|x - h| + k \)[/tex] is the point [tex]\((h, k)\)[/tex] because the value [tex]\(x = h\)[/tex] is where the expression inside the absolute value [tex]\(|x - h|\)[/tex] equals zero, and at this point, [tex]\(f(x) = k\)[/tex], which gives the minimum or maximum value of the function.
Based on this understanding, we can conclude that the vertex of the absolute value function [tex]\( f(x) = a|x - h| + k \)[/tex] is [tex]\((h, k)\)[/tex].
Therefore, the correct answer is:
[tex]\[ (h, k) \][/tex]
So, the option that represents the vertex is:
[tex]\[(h, k)\][/tex]
Answer: [tex]\((h, k)\)[/tex] (Option 4).
