- The function \(y = -2|x - 4| + 1\) is an absolute value function with a vertex at \((4, 1)\) and a downward orientation.
- Five coordinate points were identified by evaluating the function at \(x \in \{2, 3, 4, 5, 6\}\).
- The resulting points for graphing are \((4, 1), (3, -1), (5, -1), (2, -3),\) and \((6, -3)\).
- These points define a V-shaped graph with a maximum at \((4, 1)\).
Explanation
- Analyzing the Absolute Value Function
The function to be analyzed is given by \(y = -2|x - 4| + 1\). This is an absolute value function, which typically has a V-shaped graph. The expression \(|x - 4|\) indicates a horizontal shift of the standard function \(y = |x|\) by 4 units to the right. The coefficient \(-2\) indicates a vertical stretch by a factor of 2 and a reflection across the x-axis, while the constant \(+1\) indicates a vertical shift upward by 1 unit. The vertex of this function is located at the point where the expression inside the absolute value is zero, which is \(x = 4\). At this vertex, \(y = -2|4 - 4| + 1 = 1\), so the vertex is \((4, 1)\).
- Identifying Points for Graphing
To graph the function accurately, we identify points by selecting x-values around the vertex at \(x = 4\). Let us choose \(x = 2, 3, 4, 5, 6\):
- For \(x = 4\): \(y = -2|4 - 4| + 1 = -2(0) + 1 = 1\). Point: \((4, 1)\).
- For \(x = 3\): \(y = -2|3 - 4| + 1 = -2(1) + 1 = -1\). Point: \((3, -1)\).
- For \(x = 5\): \(y = -2|5 - 4| + 1 = -2(1) + 1 = -1\). Point: \((5, -1)\).
- For \(x = 2\): \(y = -2|2 - 4| + 1 = -2(2) + 1 = -3\). Point: \((2, -3)\).
- For \(x = 6\): \(y = -2|6 - 4| + 1 = -2(2) + 1 = -3\). Point: \((6, -3)\).
These five points provide the shape and orientation of the V-shaped graph.
- Finalizing the Graphing Points
With the vertex \((4, 1)\) and the calculated points \((2, -3), (3, -1), (5, -1),\) and \((6, -3)\), we can now draw the graph of the function. The graph starts at the vertex \((4, 1)\) and opens downwards, forming a V-shape. The function reaches a maximum value of \(y = 1\) at \(x = 4\) and decreases as \(x\) moves away from 4 in either direction. The identified coordinate points are \((4, 1), (3, -1), (5, -1), (2, -3), (6, -3)\), which are now clearly established to define the trajectory of the function.
Examples
Absolute value functions are often used in engineering and physics to model phenomena that behave symmetrically about a certain point. For instance, consider a bouncing ball whose height relative to time can sometimes be modeled by absolute value functions to represent its path towards and away from the ground. Understanding how to shift, stretch, and reflect these functions allows one to precisely model the movement of objects, optimize path planning for robotics, or analyze signals that possess a 'peak' or 'valley' in their intensity.