What is the vertex of the graph of [tex]g(x)=|x-8|+6[/tex]?

A. [tex](6,8)[/tex]
B. [tex](8,6)[/tex]
C. [tex](6,-8)[/tex]
D. [tex](-8,6)[/tex]

Recognize the function as an absolute value function.
Identify the general form of an absolute value function: y = ∣ x − h ∣ + k .
Determine the values of h and k from the given function: h = 8 and k = 6 .
State the vertex of the graph as ( h , k ) , which is ( 8 , 6 ) ​ .

Explanation

Understanding the Problem We are given the function g ( x ) = ∣ x − 8∣ + 6 and asked to find the vertex of its graph. The graph of an absolute value function of the form y = ∣ x − h ∣ + k is a V-shaped graph, and its vertex is located at the point ( h , k ) .

Identifying h and k Comparing the given function g ( x ) = ∣ x − 8∣ + 6 to the general form y = ∣ x − h ∣ + k , we can identify the values of h and k . In this case, h = 8 and k = 6 .

Finding the Vertex Therefore, the vertex of the graph of g ( x ) = ∣ x − 8∣ + 6 is ( 8 , 6 ) .


Examples
Imagine you're designing a skateboard ramp with a V-shaped profile. The function g ( x ) = ∣ x − 8∣ + 6 models this ramp, where the vertex (8, 6) represents the lowest point of the ramp. Knowing the vertex helps you determine the ramp's starting point and height, ensuring a smooth and safe ride. Understanding absolute value functions and their vertices is crucial in designing structures and predicting outcomes in various real-world scenarios.

Answered by GinnyAnswer | 2025-07-08