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 .
Compare the given function with the general form f ( x ) = a ∣ x − h ∣ + k to identify h and k .
The vertex of the graph is ( h , k ) .
The vertex of the graph of f ( x ) = ∣ x − 13∣ + 11 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. This is an absolute value function, which has a V-shaped graph. The general form of an absolute value function is f ( x ) = a ∣ x − h ∣ + k , where ( h , k ) is the vertex of the graph.
Identifying h and k Comparing the given function f ( x ) = ∣ x − 13∣ + 11 with the general form f ( x ) = a ∣ x − h ∣ + k , we can identify the values of h and k . In this case, a = 1 , h = 13 , and k = 11 .
Finding the Vertex Therefore, the vertex of the graph is ( h , k ) = ( 13 , 11 ) .
Examples
Absolute value functions are used in various real-world scenarios, such as determining the distance from a target or calculating deviations from a set point. For example, in manufacturing, if a machine is set to produce parts of a specific length (e.g., 13 cm), the absolute value function can model the variation in the actual length of the parts produced. If the actual length is x , then ∣ x − 13∣ represents the deviation from the target length. Adding a constant, like 11, shifts the graph upwards, which could represent a minimum acceptable deviation plus a base value. Understanding the vertex helps in analyzing the minimum deviation and the corresponding length at which it occurs.
