Sure, let's analyze each function step-by-step to determine which ones have a vertex with an [tex]\( x \)[/tex] value of 0.
### 1. [tex]\( f(x) = |x| \)[/tex]
The function [tex]\( f(x) = |x| \)[/tex] is a fundamental absolute value function. The vertex of this function is at the point where [tex]\( x = 0 \)[/tex] because it is the point where the value changes from negative to positive.
Vertex: [tex]\( (0, 0) \)[/tex]
### 2. [tex]\( f(x) = |x| + 3 \)[/tex]
The function [tex]\( f(x) = |x| + 3 \)[/tex] is the absolute value function shifted upwards by 3 units. The vertex of the original function [tex]\( |x| \)[/tex] is at [tex]\( x = 0 \)[/tex]. Shifting it upward does not affect the [tex]\( x \)[/tex]-value of the vertex.
Vertex: [tex]\( (0, 3) \)[/tex]
### 3. [tex]\( f(x) = |x + 3| \)[/tex]
The function [tex]\( f(x) = |x + 3| \)[/tex] is the absolute value function shifted to the left by 3 units. Therefore, the vertex is also shifted left from [tex]\( x = 0 \)[/tex] to [tex]\( x = -3 \)[/tex].
Vertex: [tex]\( (-3, 0) \)[/tex]
### 4. [tex]\( f(x) = |x| - 6 \)[/tex]
The function [tex]\( f(x) = |x| - 6 \)[/tex] is the absolute value function shifted downward by 6 units. The vertex of [tex]\( |x| \)[/tex] remains at [tex]\( x = 0 \)[/tex], but the [tex]\( y \)[/tex]-value decreases by 6.
Vertex: [tex]\( (0, -6) \)[/tex]
### 5. [tex]\( f(x) = |x + 3| - 6 \)[/tex]
The function [tex]\( f(x) = |x + 3| - 6 \)[/tex] is the function [tex]\( |x + 3| \)[/tex], which we determined has a vertex at [tex]\( x = -3 \)[/tex], shifted downward by 6 units. Therefore, the vertex is at [tex]\( x = -3 \)[/tex].
Vertex: [tex]\( (-3, -6) \)[/tex]
Now, we need to select the functions that have a vertex with an [tex]\( x \)[/tex] value of 0. From our analysis, these functions are:
- [tex]\( f(x) = |x| \)[/tex]
- [tex]\( f(x) = |x| + 3 \)[/tex]
- [tex]\( f(x) = |x| - 6 \)[/tex]
Therefore, the correct options are:
1, 2, and 4.
