Verified Answer :

  • Analyze the given equation \(y = -2|x - 7| + 2\) as an absolute value function in the vertex form \(y = a|x - h| + k\).
  • Identify the parameters \(h = 7\) and \(k = 2\) by comparing the equation to the standard form.
  • Conclude that the vertex \((h, k)\) is \((7, 2)\).
  • Select the option \((7, 2)\) as the correct answer, which is \(\boxed{(7, 2)}\).

Explanation

  1. Analyzing the Function and Identifying the Vertex Form

The given equation is \(y = -2|x - 7| + 2\). This is an absolute value function, which generally follows the vertex form \(y = a|x - h| + k\), where \((h, k)\) represents the coordinates of the vertex of the function's graph. Identifying the vertex allows us to understand the peak or turning point of the graph, which is fundamental in analyzing functions of this type.

  1. Mapping the Components to Identify the Vertex

To find the vertex, we compare the given equation \(y = -2|x - 7| + 2\) directly to the standard vertex form \(y = a|x - h| + k\). By inspection:

  • The value of \(a\) is \(-2\).
  • The value of \(h\) is \(7\).
  • The value of \(k\) is \(2\).

Therefore, the vertex of the function \((h, k)\) is \((7, 2)\).

  1. Conclusion and Selection of the Correct Option

Comparing our identified vertex \((7, 2)\) to the provided options, we see that it matches the second option. Thus, the vertex of the function \(y = -2|x - 7| + 2\) is \((7, 2)\). The final answer is \(\boxed{(7, 2)}\).

Examples

The concept of identifying the vertex of an absolute value function is very useful in engineering and physics, particularly when modeling pathways that change direction sharply. For example, if a laser beam bounces off a surface at a point, the path of the light can often be represented by an absolute value function. Finding the vertex allows engineers to calculate exactly where the bounce occurs, ensuring precision in optical systems or sound reflection designs.