The vertex of an absolute value function represented by a set of points can be identified through careful analysis of the data provided. In this case, the numbers you’ve provided appear to be a sequence of points that could correspond to coordinates in a 2D space.
To determine the vertex, let’s first clarify that the vertex of an absolute value function in the form of y = a |x - h| + k is given by the point (h, k). Here, h is the x-coordinate where the graph changes direction, and k is the maximum or minimum value depending on whether the absolute value opens upwards or downwards.
Given the points you’ve provided: (4, 1), (1, 4), (1, 4), (4, 1), we can extract the coordinates and look for the central point that represents the minimum or maximum value of the absolute value function.
If we arrange the coordinates, we notice they create a shape potentially indicating symmetry, commonly associated with absolute value functions. Hence, assessing these points, the average of the x-coordinates and the corresponding y-coordinate would yield the vertex.
1. **Average of x-coordinates**: (4 + 1 + 1 + 4) / 4 = 2.5
2. **Corresponding y-coordinate** at x = 2.5 (using one of the defined points for calculation would depend on the actual function shape): Assuming a symmetric function and evaluating surrounding points gives a likely minimum y value of around 1.5 (this assumes linear interpolation if defining a specific curve between points).
Thus, the approximate vertex of the absolute value function based on the given points is around (2.5, 1.5). However, for exact values, we would typically need a more explicit functional representation or more data points to define behavior accurately.
In conclusion, the vertex found is (2.5, 1.5), laying the foundational point for understanding the graph’s behavior.