Which functions have a vertex with an [tex]$x$[/tex]-value of 0? Select three options.

[tex]$f(x)=|x|$[/tex]
[tex]$f(x)=|x|+3$[/tex]
[tex]$f(x)=|x+3|$[/tex]
[tex]$f(x)=|x|-6$[/tex]
[tex]$f(x)=|x+3|-6$[/tex]

The functions with a vertex x-value of 0 are f ( x ) = ∣ x ∣ , f ( x ) = ∣ x ∣ + 3 , and f ( x ) = ∣ x ∣ − 6 . These functions maintain their vertex at the point (0, k). Therefore, the answer includes the first, second, and fourth options from the list provided.
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Answered by Anonymous | 2025-07-08

The vertex of an absolute value function f ( x ) = a ∣ x − h ∣ + k is (h, k).
Identify the vertex of each given function.
Select the functions where the x-value of the vertex (h) is 0.
The functions with a vertex x-value of 0 are: f ( x ) = ∣ x ∣ , f ( x ) = ∣ x ∣ + 3 , and f ( x ) = ∣ x ∣ − 6 .

Explanation

Understanding the Problem We need to identify which of the given absolute value functions have a vertex with an x-value of 0. 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, we are looking for functions where h = 0 .

Analyzing Each Function Let's analyze each function:

f ( x ) = ∣ x ∣ . This can be written as f ( x ) = ∣ x − 0∣ + 0 . The vertex is (0, 0), so the x-value of the vertex is 0.

f ( x ) = ∣ x ∣ + 3 . This can be written as f ( x ) = ∣ x − 0∣ + 3 . The vertex is (0, 3), so the x-value of the vertex is 0.

f ( x ) = ∣ x + 3∣ . This can be written as f ( x ) = ∣ x − ( − 3 ) ∣ + 0 . The vertex is (-3, 0), so the x-value of the vertex is -3.

f ( x ) = ∣ x ∣ − 6 . This can be written as f ( x ) = ∣ x − 0∣ − 6 . The vertex is (0, -6), so the x-value of the vertex is 0.

f ( x ) = ∣ x + 3∣ − 6 . This can be written as f ( x ) = ∣ x − ( − 3 ) ∣ − 6 . The vertex is (-3, -6), so the x-value of the vertex is -3.

Identifying the Functions The functions with a vertex x-value of 0 are:



f ( x ) = ∣ x ∣
f ( x ) = ∣ x ∣ + 3
f ( x ) = ∣ x ∣ − 6

Examples
Absolute value functions are used in various real-world scenarios, such as calculating distances or deviations from a target value. For example, in manufacturing, if a machine is set to produce parts with a specific length, the absolute value function can be used to model the difference between the actual length of a part and the target length. If the target length is 5 cm, the function f ( x ) = ∣ x − 5∣ represents the deviation from the target, where x is the actual length. The vertex of this function, (5, 0), indicates that the minimum deviation is 0 when the part's length is exactly 5 cm.

Answered by GinnyAnswer | 2025-07-08