Find the vertex of the given function. [tex]f(x)=|x-5|+10[/tex] The vertex is at ($\square$, $\square$).

The vertex of the function f ( x ) = ∣ x − 5∣ + 10 is at the point ( 5 , 10 ) . This is identified by recognizing the function's parameters in the absolute value form. Therefore, the coordinates of the vertex are ( 5 , 10 ) ​ .
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Answered by Anonymous | 2025-07-04

The given function is an absolute value function in the form f ( x ) = ∣ x − 5∣ + 10 .
The general form of an absolute value function is f ( x ) = a ∣ x − h ∣ + k , where the vertex is at ( h , k ) .
Identify h and k from the given function: h = 5 and k = 10 .
The vertex of the function is ( 5 , 10 ) ​ .

Explanation

Understanding the Problem The given function is f ( x ) = ∣ x − 5∣ + 10 . We need to find the vertex of this function. The function is an absolute value function.

Identifying the Vertex Form The general form of an absolute value function is f ( x ) = a ∣ x − h ∣ + k , where the vertex is at ( h , k ) . In our case, a = 1 , h = 5 , and k = 10 .

Finding the Vertex Therefore, the vertex of the function f ( x ) = ∣ x − 5∣ + 10 is ( 5 , 10 ) .


Examples
Absolute value functions are used in various real-world scenarios, such as determining the distance from a target or setting tolerance levels in manufacturing. For example, if a machine needs to cut a metal rod to 5 cm with a tolerance of 0.1 cm, the length x of the rod must satisfy ∣ x − 5∣ ≤ 0.1 . This ensures that the rod is within the acceptable length range. Understanding absolute value functions helps in quality control and precision measurements.

Answered by GinnyAnswer | 2025-07-04