What is the vertex of the graph of $f(x)=|x+5|-6$?

The vertex of the graph of the function f ( x ) = ∣ x + 5∣ − 6 is located at the point ( − 5 , − 6 ) . This point is found by setting the inside of the absolute value to zero and then substituting back into the function to find the y-value. Thus, the complete vertex is ( − 5 , − 6 ) .
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Answered by Anonymous | 2025-07-03

Set the expression inside the absolute value to zero: x + 5 = 0 .
Solve for x : x = − 5 .
Substitute x = − 5 into the function: f ( − 5 ) = ∣0∣ − 6 = − 6 .
The vertex of the graph is ( − 5 , − 6 ) ​ .

Explanation

Understanding the Problem We are given the function f ( x ) = ∣ x + 5∣ − 6 and asked to find the vertex of its graph. The absolute value function ∣ x ∣ has a vertex at x = 0 . Similarly, ∣ x + 5∣ has a vertex when x + 5 = 0 .

Finding the x-coordinate To find the x-coordinate of the vertex, we set the expression inside the absolute value to zero and solve for x: x + 5 = 0 x = − 5

Finding the y-coordinate Now we substitute x = − 5 into the function to find the y-coordinate of the vertex: f ( − 5 ) = ∣ − 5 + 5∣ − 6 = ∣0∣ − 6 = 0 − 6 = − 6

Stating the Vertex Therefore, the vertex of the graph of f ( x ) = ∣ x + 5∣ − 6 is ( − 5 , − 6 ) .


Examples
Understanding the vertex of an absolute value function is useful in various real-world scenarios. For example, in physics, if you're analyzing the trajectory of a bouncing ball, the vertex represents the highest point the ball reaches. Similarly, in business, if you're modeling profit as a function of investment, the vertex could represent the point of maximum profit. Knowing how to find the vertex helps in optimizing these situations.

Answered by GinnyAnswer | 2025-07-03