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 (2)
The given function is f ( x ) = ∣ x − 13∣ + 11 .
Compare the given function with the general form f ( x ) = a ∣ x − h ∣ + k .
Identify the values: h = 13 and k = 11 .
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. This is an absolute value function, which has a V-shape. 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 the Parameters 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 a , h , and k . In this case, a = 1 , h = 13 , and k = 11 .
Finding the Vertex The vertex of the graph is ( h , k ) = ( 13 , 11 ) .
Examples
Absolute value functions are used in many real-world applications, such as finding the distance between two points or modeling situations where only the magnitude of a value is important. For example, in manufacturing, the tolerance of a part can be expressed as an absolute value function, where the ideal measurement is the vertex and the tolerance range is the spread of the V-shape. Understanding how to find the vertex helps engineers ensure that parts meet the required specifications. Another example is in navigation, where the absolute value function can be used to model the distance from a specific location, regardless of direction.
The vertex of the graph of the function f ( x ) = ∣ x − 13∣ + 11 is ( 13 , 11 ) , which corresponds to option D. This point represents the minimum value of the function, where the V-shaped graph reaches its lowest point.
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