Find the vertex of the graphed function. [tex]f(x)=|x-4|+3[/tex]
The vertex is at ( $\square$ , $\square$ ).

The vertex of the function f ( x ) = ∣ x − 4∣ + 3 is found by identifying the values of h and k in the standard form of absolute value functions, which gives the vertex as ( 4 , 3 ) .
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Answered by Anonymous | 2025-07-04

The function is in the form f ( x ) = ∣ x − h ∣ + k .
Identify h and k from the given function f ( x ) = ∣ x − 4∣ + 3 .
The vertex is at ( h , k ) .
The vertex of the function is ( 4 , 3 ) ​ .

Explanation

Understanding the Function The given function is f ( x ) = ∣ x − 4∣ + 3 . We need to find the vertex of this absolute value function.

Identifying h and k The general form of an absolute value function is f ( x ) = a ∣ x − h ∣ + k , where ( h , k ) represents the vertex of the function. In our case, f ( x ) = ∣ x − 4∣ + 3 , we can identify h and k .

Determining the Vertex Comparing f ( x ) = ∣ x − 4∣ + 3 with the general form f ( x ) = ∣ x − h ∣ + k , we can see that h = 4 and k = 3 . Therefore, the vertex of the function is at the point ( 4 , 3 ) .

Final Answer The vertex of the function f ( x ) = ∣ x − 4∣ + 3 is ( 4 , 3 ) .


Examples
Absolute value functions are used in various real-world applications, such as determining the distance from a target in engineering or physics. For example, if you are designing a robot that needs to move to a specific point, the absolute value function can help you calculate the distance the robot needs to travel, regardless of direction. Similarly, in finance, absolute value functions can be used to model deviations from a target value, such as tracking how much an investment portfolio deviates from its expected return. Understanding the vertex of an absolute value function helps in optimizing these processes by identifying the point of minimum deviation or distance.

Answered by GinnyAnswer | 2025-07-04