What is the vertex of the graph of $f(x)=|x-13|+11$?
A. $(-11,13)$
B. $(-13,11)$
C. $(11,13)$
D. $(13,11)$
Answer (1)
The given function is f ( x ) = ∣ x − 13∣ + 11 .
The vertex of the absolute value function ∣ x − h ∣ + k is ( h , k ) .
Identify h = 13 and k = 11 from the given function.
The vertex of the graph is ( 13 , 11 ) .
Explanation
Understanding the Problem We are given the function f ( x ) = ∣ x − 13∣ + 11 and asked to find the vertex of its graph. The absolute value function creates a V-shaped graph, and the vertex is the point where the graph changes direction.
General Form of Absolute Value Function The general form of an absolute value function is f ( x ) = ∣ x − h ∣ + k , where the vertex of the graph is at the point ( h , k ) .
Identifying h and k In our case, we have f ( x ) = ∣ x − 13∣ + 11 . Comparing this to the general form, we can identify h = 13 and k = 11 .
Finding the Vertex Therefore, the vertex of the graph of f ( x ) = ∣ x − 13∣ + 11 is ( 13 , 11 ) .
Examples
Understanding the vertex of an absolute value function is useful in various real-world scenarios. For example, consider a manufacturing process where the deviation from a target value is equally costly whether it's above or below the target. The absolute value function models this cost, and the vertex represents the point of minimum cost, which is the target value. Knowing the vertex helps optimize the process to minimize deviations and costs. Another example is in physics, where absolute value functions can model distances, and the vertex can represent a point of reference or equilibrium.
