What is the vertex of the graph of $f(x)=|x+3|+7$?
A. $(3,7)$
B. $(7,3)$
C. $(-3,7)$
D. $(7,-3)$
Answer (1)
The function is in the form f ( x ) = ∣ x + 3∣ + 7 .
Rewrite the function as f ( x ) = ∣ x − ( − 3 ) ∣ + 7 to match the vertex form f ( x ) = a ∣ x − h ∣ + k .
Identify the vertex coordinates as ( h , k ) = ( − 3 , 7 ) .
The vertex of the graph is ( − 3 , 7 ) .
Explanation
Understanding the Function The given function is f ( x ) = ∣ x + 3∣ + 7 . We need to find the vertex of the graph of this absolute value function.
Identifying the Vertex Form The general form of an absolute value function is f ( x ) = a ∣ x − h ∣ + k , where ( h , k ) is the vertex of the graph. In our case, we can rewrite the given function as f ( x ) = ∣ x − ( − 3 ) ∣ + 7 .
Determining the Vertex Coordinates Comparing f ( x ) = ∣ x − ( − 3 ) ∣ + 7 with the general form f ( x ) = a ∣ x − h ∣ + k , we can identify h = − 3 and k = 7 . Therefore, the vertex of the graph is ( − 3 , 7 ) .
Final Answer The vertex of the graph of f ( x ) = ∣ x + 3∣ + 7 is ( − 3 , 7 ) .
Examples
Absolute value functions are used in various real-world scenarios, such as determining the distance from a target or modeling situations where only the magnitude of a value matters. For example, in engineering, when designing a bridge, the absolute value function can be used to model the maximum stress a material can withstand, regardless of whether the stress is tensile or compressive. Similarly, in finance, it can be used to model the absolute change in stock prices, focusing on the magnitude of the change rather than the direction. Understanding the vertex of an absolute value function helps in optimizing these models to find the minimum or maximum values in these scenarios.
