To determine which functions have a vertex with an [tex]\( x \)[/tex]-value of 0, we need to closely examine the behavior of absolute value functions.
Let's break down each function:
1. Function: [tex]\( f(x) = |x| \)[/tex]
- The absolute value function [tex]\( |x| \)[/tex] has its vertex at [tex]\( x = 0 \)[/tex]. This is because [tex]\( |x| \)[/tex] reaches its minimum value of 0 when [tex]\( x = 0 \)[/tex]. So, this function has a vertex at [tex]\( x = 0 \)[/tex].
2. Function: [tex]\( f(x) = |x| + 3 \)[/tex]
- Adding 3 to [tex]\( |x| \)[/tex] shifts the graph upwards but does not change the [tex]\( x \)[/tex]-coordinate of the vertex. Hence, the vertex is still at [tex]\( x = 0 \)[/tex].
3. Function: [tex]\( f(x) = |x + 3| \)[/tex]
- This function shifts the graph of [tex]\( |x| \)[/tex] to the left by 3 units. The vertex for this function is at [tex]\( x = -3 \)[/tex], not at [tex]\( x = 0 \)[/tex].
4. Function: [tex]\( f(x) = |x| - 6 \)[/tex]
- Subtracting 6 from [tex]\( |x| \)[/tex] shifts the graph downwards but does not change the [tex]\( x \)[/tex]-coordinate of the vertex. Thus, the vertex remains at [tex]\( x = 0 \)[/tex].
5. Function: [tex]\( f(x) = |x + 3| - 6 \)[/tex]
- First, [tex]\( |x + 3| \)[/tex] shifts the graph of [tex]\( |x| \)[/tex] to the left by 3 units, making the vertex [tex]\( x = -3 \)[/tex]. Subtracting 6 shifts the graph downwards but does not affect the [tex]\( x \)[/tex]-coordinate of the vertex, which still is [tex]\( x = -3 \)[/tex].
Thus, the functions that have a vertex with an [tex]\( x \)[/tex]-value of 0 are:
- [tex]\( f(x) = |x| \)[/tex]
- [tex]\( f(x) = |x| + 3 \)[/tex]
- [tex]\( f(x) = |x| - 6 \)[/tex]
These are the functions whose vertices lie on the [tex]\( x = 0 \)[/tex] axis.
